A036040 -id:A036040 - cp's OEIS frontend

A000110 Bell or exponential numbers: number of ways to partition a set of n labeled elements.

1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, 115975, 678570, 4213597, 27644437, 190899322, 1382958545, 10480142147, 82864869804, 682076806159, 5832742205057, 51724158235372, 474869816156751, 4506715738447323, 44152005855084346, 445958869294805289, 4638590332229999353, 49631246523618756274

Offset: 0

N. J. A. Sloane

The leading diagonal of its difference table is the sequence shifted, see Bernstein and Sloane (1995). - N. J. A. Sloane, Jul 04 2015
Also the number of equivalence relations that can be defined on a set of n elements. - Federico Arboleda (federico.arboleda(AT)gmail.com), Mar 09 2005
a(n) = number of nonisomorphic colorings of a map consisting of a row of n+1 adjacent regions. Adjacent regions cannot have the same color. - David W. Wilson, Feb 22 2005
If an integer is squarefree and has n distinct prime factors then a(n) is the number of ways of writing it as a product of its divisors. - Amarnath Murthy, Apr 23 2001
Consider rooted trees of height at most 2. Letting each tree 'grow' into the next generation of n means we produce a new tree for every node which is either the root or at height 1, which gives the Bell numbers. - Jon Perry, Jul 23 2003
Begin with [1,1] and follow the rule that [1,k] -> [1,k+1] and [1,k] k times, e.g., [1,3] is transformed to [1,4], [1,3], [1,3], [1,3]. Then a(n) is the sum of all components: [1,1] = 2; [1,2], [1,1] = 5; [1,3], [1,2], [1,2], [1,2], [1,1] = 15; etc. - Jon Perry, Mar 05 2004
Number of distinct rhyme schemes for a poem of n lines: a rhyme scheme is a string of letters (e.g., 'abba') such that the leftmost letter is always 'a' and no letter may be greater than one more than the greatest letter to its left. Thus 'aac' is not valid since 'c' is more than one greater than 'a'. For example, a(3)=5 because there are 5 rhyme schemes: aaa, aab, aba, abb, abc; also see example by Neven Juric. - Bill Blewett, Mar 23 2004
In other words, number of length-n restricted growth strings (RGS) [s(0),s(1),...,s(n-1)] where s(0)=0 and s(k) <= 1 + max(prefix) for k >= 1, see example (cf. A080337 and A189845). - Joerg Arndt, Apr 30 2011
Number of partitions of {1, ..., n+1} into subsets of nonconsecutive integers, including the partition 1|2|...|n+1. E.g., a(3)=5: there are 5 partitions of {1,2,3,4} into subsets of nonconsecutive integers, namely, 13|24, 13|2|4, 14|2|3, 1|24|3, 1|2|3|4. - Augustine O. Munagi, Mar 20 2005
Triangle (addition) scheme to produce terms, derived from the recurrence, from Oscar Arevalo (loarevalo(AT)sbcglobal.net), May 11 2005:
    1
    1  2
    2  3  5
    5  7 10 15
   15 20 27 37 52
   ... [This is Aitken's array A011971]
With P(n) = the number of integer partitions of n, p(i) = the number of parts of the i-th partition of n, d(i) = the number of different parts of the i-th partition of n, p(j,i) = the j-th part of the i-th partition of n, m(i,j) = multiplicity of the j-th part of the i-th partition of n, one has: a(n) = Sum_{i=1..P(n)} (n!/(Product_{j=1..p(i)} p(i,j)!)) * (1/(Product_{j=1..d(i)} m(i,j)!)). - Thomas Wieder, May 18 2005
a(n+1) is the number of binary relations on an n-element set that are both symmetric and transitive. - Justin Witt (justinmwitt(AT)gmail.com), Jul 12 2005
If the rule from Jon Perry, Mar 05 2004, is used, then a(n-1) = [number of components used to form a(n)] / 2. - Daniel Kuan (dkcm(AT)yahoo.com), Feb 19 2006
a(n) is the number of functions f from {1,...,n} to {1,...,n,n+1} that satisfy the following two conditions for all x in the domain: (1) f(x) > x; (2) f(x)=n+1 or f(f(x))=n+1. E.g., a(3)=5 because there are exactly five functions that satisfy the two conditions: f1={(1,4),(2,4),(3,4)}, f2={(1,4),(2,3),(3,4)}, f3={(1,3),(2,4),(3,4)}, f4={(1,2),(2,4),(3,4)} and f5={(1,3),(2,3),(3,4)}. - Dennis P. Walsh, Feb 20 2006
Number of asynchronic siteswap patterns of length n which have no zero-throws (i.e., contain no 0's) and whose number of orbits (in the sense given by Allen Knutson) is equal to the number of balls. E.g., for n=4, the condition is satisfied by the following 15 siteswaps: 4444, 4413, 4242, 4134, 4112, 3441, 2424, 1344, 2411, 1313, 1241, 2222, 3131, 1124, 1111. Also number of ways to choose n permutations from identity and cyclic permutations (1 2), (1 2 3), ..., (1 2 3 ... n) so that their composition is identity. For n=3 we get the following five: id o id o id, id o (1 2) o (1 2), (1 2) o id o (1 2), (1 2) o (1 2) o id, (1 2 3) o (1 2 3) o (1 2 3). (To see the bijection, look at Ehrenborg and Readdy paper.) - Antti Karttunen, May 01 2006
a(n) is the number of permutations on [n] in which a 3-2-1 (scattered) pattern occurs only as part of a 3-2-4-1 pattern. Example: a(3) = 5 counts all permutations on [3] except 321. See "Eigensequence for Composition" reference a(n) = number of permutation tableaux of size n (A000142) whose first row contains no 0's. Example: a(3)=5 counts {{}, {}, {}}, {{1}, {}}, {{1}, {0}}, {{1}, {1}}, {{1, 1}}. - David Callan, Oct 07 2006
From Gottfried Helms, Mar 30 2007: (Start)
This sequence is also the first column in the matrix-exponential of the (lower triangular) Pascal-matrix, scaled by exp(-1): PE = exp(P) / exp(1) =
     1
     1    1
     2    2    1
     5    6    3    1
    15   20   12    4    1
    52   75   50   20    5    1
   203  312  225  100   30    6   1
   877 1421 1092  525  175   42   7   1
First 4 columns are A000110, A033306, A105479, A105480. The general case is mentioned in the two latter entries. PE is also the Hadamard-product Toeplitz(A000110) (X) P:
     1
     1    1
     2    1    1
     5    2    1    1
    15    5    2    1    1      (X) P
    52   15    5    2    1    1
   203   52   15    5    2    1   1
   877  203   52   15    5    2   1   1
(End)
The terms can also be computed with finite steps and precise integer arithmetic. Instead of exp(P)/exp(1) one can compute A = exp(P - I) where I is the identity-matrix of appropriate dimension since (P-I) is nilpotent to the order of its dimension. Then a(n)=A[n,1] where n is the row-index starting at 1. - Gottfried Helms, Apr 10 2007
When n is prime, a(n) == 2 (mod n), but the converse is not always true. Define a Bell pseudoprime to be a composite number n such that a(n) == 2 (mod n). W. F. Lunnon recently found the Bell pseudoprimes 21361 = 41*521 and C46 = 3*23*16218646893090134590535390526854205539989357 and conjectured that Bell pseudoprimes are extremely scarce. So the second Bell pseudoprime is unlikely to be known with certainty in the near future. I confirmed that 21361 is the first. - David W. Wilson, Aug 04 2007 and Sep 24 2007
This sequence and A000587 form a reciprocal pair under the list partition transform described in A133314. - Tom Copeland, Oct 21 2007
Starting (1, 2, 5, 15, 52, ...), equals row sums and right border of triangle A136789. Also row sums of triangle A136790. - Gary W. Adamson, Jan 21 2008
This is the exponential transform of A000012. - Thomas Wieder, Sep 09 2008
From Abdullahi Umar, Oct 12 2008: (Start)
a(n) is also the number of idempotent order-decreasing full transformations (of an n-chain).
a(n) is also the number of nilpotent partial one-one order-decreasing transformations (of an n-chain).
a(n+1) is also the number of partial one-one order-decreasing transformations (of an n-chain). (End)
From Peter Bala, Oct 19 2008: (Start)
Bell(n) is the number of n-pattern sequences [Cooper & Kennedy]. An n-pattern sequence is a sequence of integers (a_1,...,a_n) such that a_i = i or a_i = a_j for some j < i. For example, Bell(3) = 5 since the 3-pattern sequences are (1,1,1), (1,1,3), (1,2,1), (1,2,2) and (1,2,3).
Bell(n) is the number of sequences of positive integers (N_1,...,N_n) of length n such that N_1 = 1 and N_(i+1) <= 1 + max{j = 1..i} N_j for i >= 1 (see the comment by B. Blewett above). It is interesting to note that if we strengthen the latter condition to N_(i+1) <= 1 + N_i we get the Catalan numbers A000108 instead of the Bell numbers.
(End)
Equals the eigensequence of Pascal's triangle, A007318; and starting with offset 1, = row sums of triangles A074664 and A152431. - Gary W. Adamson, Dec 04 2008
The entries f(i, j) in the exponential of the infinite lower-triangular matrix of binomial coefficients b(i, j) are f(i, j) = b(i, j) e a(i - j). - David Pasino, Dec 04 2008
Equals lim_{k->oo} A071919^k. - Gary W. Adamson, Jan 02 2009
Equals A154107 convolved with A014182, where A014182 = expansion of exp(1-x-exp(-x)), the eigensequence of A007318^(-1). Starting with offset 1 = A154108 convolved with (1,2,3,...) = row sums of triangle A154109. - Gary W. Adamson, Jan 04 2009
Repeated iterates of (binomial transform of [1,0,0,0,...]) will converge upon (1, 2, 5, 15, 52, ...) when each result is prefaced with a "1"; such that the final result is the fixed limit: (binomial transform of [1,1,2,5,15,...]) = (1,2,5,15,52,...). - Gary W. Adamson, Jan 14 2009
From Karol A. Penson, May 03 2009: (Start)
Relation between the Bell numbers B(n) and the n-th derivative of 1/Gamma(1+x) evaluated at x=1:
a) produce a number of such derivatives through seq(subs(x=1, simplify((d^n/dx^n)GAMMA(1+x)^(-1))), n=1..5);
b) leave them expressed in terms of digamma (Psi(k)) and polygamma (Psi(k,n)) functions and unevaluated;
Examples of such expressions, for n=1..5, are:
n=1: -Psi(1),
n=2: -(-Psi(1)^2 + Psi(1,1)),
n=3: -Psi(1)^3 + 3*Psi(1)*Psi(1,1) - Psi(2,1),
n=4: -(-Psi(1)^4 + 6*Psi(1)^2*Psi(1,1) - 3*Psi(1,1)^2 - 4*Psi(1)*Psi(2,1) + Psi(3, 1)),
n=5: -Psi(1)^5 + 10*Psi(1)^3*Psi(1,1) - 15*Psi(1)*Psi(1,1)^2 - 10*Psi(1)^2*Psi(2,1) + 10*Psi(1,1)*Psi(2,1) + 5*Psi(1)*Psi(3,1) - Psi(4,1);
c) for a given n, read off the sum of absolute values of coefficients of every term involving digamma or polygamma functions.
This sum is equal to B(n). Examples: B(1)=1, B(2)=1+1=2, B(3)=1+3+1=5, B(4)=1+6+3+4+1=15, B(5)=1+10+15+10+10+5+1=52;
d) Observe that this decomposition of the Bell number B(n) apparently does not involve the Stirling numbers of the second kind explicitly.
(End)
The numbers given above by Penson lead to the multinomial coefficients A036040. - Johannes W. Meijer, Aug 14 2009
Column 1 of A162663. - Franklin T. Adams-Watters, Jul 09 2009
Asymptotic expansions (0!+1!+2!+...+(n-1)!)/(n-1)! = a(0) + a(1)/n + a(2)/n^2 + ... and (0!+1!+2!+...+n!)/n! = 1 + a(0)/n + a(1)/n^2 + a(2)/n^3 + .... - Michael Somos, Jun 28 2009
Starting with offset 1 = row sums of triangle A165194. - Gary W. Adamson, Sep 06 2009
a(n+1) = A165196(2^n); where A165196 begins: (1, 2, 4, 5, 7, 12, 14, 15, ...). such that A165196(2^3) = 15 = A000110(4). - Gary W. Adamson, Sep 06 2009
The divergent series g(x=1,m) = 1^m*1! - 2^m*2! + 3^m*3! - 4^m*4! + ..., m >= -1, which for m=-1 dates back to Euler, is related to the Bell numbers. We discovered that g(x=1,m) = (-1)^m * (A040027(m) - A000110(m+1) * A073003). We observe that A073003 is Gompertz's constant and that A040027 was published by Gould, see for more information A163940. - Johannes W. Meijer, Oct 16 2009
a(n) = E(X^n), i.e., the n-th moment about the origin of a random variable X that has a Poisson distribution with (rate) parameter, lambda = 1. - Geoffrey Critzer, Nov 30 2009
Let A000110 = S(x), then S(x) = A(x)/A(x^2) when A(x) = A173110; or (1, 1, 2, 5, 15, 52, ...) = (1, 1, 3, 6, 20, 60, ...) / (1, 0, 1, 0, 3, 0, 6, 0, 20, ...). - Gary W. Adamson, Feb 09 2010
The Bell numbers serve as the upper limit for the number of distinct homomorphic images from any given finite universal algebra. Every algebra homomorphism is determined by its kernel, which must be a congruence relation. As the number of possible congruence relations with respect to a finite universal algebra must be a subset of its possible equivalence classes (given by the Bell numbers), it follows naturally. - Max Sills, Jun 01 2010
For a proof of the o.g.f. given in the R. Stephan comment see, e.g., the W. Lang link under A071919. - Wolfdieter Lang, Jun 23 2010
Let B(x) = (1 + x + 2x^2 + 5x^3 + ...). Then B(x) is satisfied by A(x)/A(x^2) where A(x) = polcoeff A173110: (1 + x + 3x^2 + 6x^3 + 20x^4 + 60x^5 + ...) = B(x) * B(x^2) * B(x^4) * B(x^8) * .... - Gary W. Adamson, Jul 08 2010
Consider a set of A000217(n) balls of n colors in which, for each integer k = 1 to n, exactly one color appears in the set a total of k times. (Each ball has exactly one color and is indistinguishable from other balls of the same color.) a(n+1) equals the number of ways to choose 0 or more balls of each color without choosing any two colors the same positive number of times. (See related comments for A000108, A008277, A016098.) - Matthew Vandermast, Nov 22 2010
A binary counter with faulty bits starts at value 0 and attempts to increment by 1 at each step. Each bit that should toggle may or may not do so. a(n) is the number of ways that the counter can have the value 0 after n steps. E.g., for n=3, the 5 trajectories are 0,0,0,0; 0,1,0,0; 0,1,1,0; 0,0,1,0; 0,1,3,0. - David Scambler, Jan 24 2011
No Bell number is divisible by 8, and no Bell number is congruent to 6 modulo 8; see Theorem 6.4 and Table 1.7 in Lunnon, Pleasants and Stephens. - Jon Perry, Feb 07 2011, clarified by Eric Rowland, Mar 26 2014
a(n+1) is the number of (symmetric) positive semidefinite n X n 0-1 matrices. These correspond to equivalence relations on {1,...,n+1}, where matrix element M[i,j] = 1 if and only if i and j are equivalent to each other but not to n+1. - Robert Israel, Mar 16 2011
a(n) is the number of monotonic-labeled forests on n vertices with rooted trees of height less than 2. We note that a labeled rooted tree is monotonic-labeled if the label of any parent vertex is greater than the label of any offspring vertex. See link "Counting forests with Stirling and Bell numbers". - Dennis P. Walsh, Nov 11 2011
a(n) = D^n(exp(x)) evaluated at x = 0, where D is the operator (1+x)*d/dx. Cf. A000772 and A094198. - Peter Bala, Nov 25 2011
B(n) counts the length n+1 rhyme schemes without repetitions. E.g., for n=2 there are 5 rhyme schemes of length 3 (aaa, aab, aba, abb, abc), and the 2 without repetitions are aba, abc. This is basically O. Munagi's result that the Bell numbers count partitions into subsets of nonconsecutive integers (see comment above dated Mar 20 2005). - Eric Bach, Jan 13 2012
Right and left borders and row sums of A212431 = A000110 or a shifted variant. - Gary W. Adamson, Jun 21 2012
Number of maps f: [n] -> [n] where f(x) <= x and f(f(x)) = f(x) (projections). - Joerg Arndt, Jan 04 2013
Permutations of [n] avoiding any given one of the 8 dashed patterns in the equivalence classes (i) 1-23, 3-21, 12-3, 32-1, and (ii) 1-32, 3-12, 21-3, 23-1. (See Claesson 2001 reference.) - David Callan, Oct 03 2013
Conjecture: No a(n) has the form x^m with m > 1 and x > 1. - Zhi-Wei Sun, Dec 02 2013
Sum_{n>=0} a(n)/n! = e^(e-1) = 5.57494152476..., see A234473. - Richard R. Forberg, Dec 26 2013 (This is the e.g.f. for x=1. - Wolfdieter Lang, Feb 02 2015)
Sum_{j=0..n} binomial(n,j)*a(j) = (1/e)*Sum_{k>=0} (k+1)^n/k! = (1/e) Sum_{k=1..oo} k^(n+1)/k! = a(n+1), n >= 0, using the Dobinski formula. See the comment by Gary W. Adamson, Dec 04 2008 on the Pascal eigensequence. - Wolfdieter Lang, Feb 02 2015
In fact it is not really an eigensequence of the Pascal matrix; rather the Pascal matrix acts on the sequence as a shift. It is an eigensequence (the unique eigensequence with eigenvalue 1) of the matrix derived from the Pascal matrix by adding at the top the row [1, 0, 0, 0 ...]. The binomial sum formula may be derived from the definition in terms of partitions: label any element X of a set S of N elements, and let X(k) be the number of subsets of S containing X with k elements. Since each subset has a unique coset, the number of partitions p(N) of S is given by p(N) = Sum_{k=1..N} (X(k) p(N-k)); trivially X(k) = N-1 choose k-1. - Mason Bogue, Mar 20 2015
a(n) is the number of ways to nest n matryoshkas (Russian nesting dolls): we may identify {1, 2, ..., n} with dolls of ascending sizes and the sets of a set partition with stacks of dolls. - Carlo Sanna, Oct 17 2015
Number of permutations of [n] where the initial elements of consecutive runs of increasing elements are in decreasing order. a(4) = 15: `1234, `2`134, `23`14, `234`1, `24`13, `3`124, `3`2`14, `3`24`1, `34`12, `34`2`1, `4`123, `4`2`13, `4`23`1, `4`3`12, `4`3`2`1. - Alois P. Heinz, Apr 27 2016
Taking with alternating signs, the Bell numbers are the coefficients in the asymptotic expansion (Ramanujan): (-1)^n*(A000166(n) - n!/exp(1)) ~ 1/n - 2/n^2 + 5/n^3 - 15/n^4 + 52/n^5 - 203/n^6 + O(1/n^7). - Vladimir Reshetnikov, Nov 10 2016
Number of treeshelves avoiding pattern T231. See A278677 for definitions and examples. - Sergey Kirgizov, Dec 24 2016
Presumably this satisfies Benford's law, although the results in Hürlimann (2009) do not make this clear. - N. J. A. Sloane, Feb 09 2017
a(n) = Sum(# of standard immaculate tableaux of shape m, m is a composition of n), where this sum is over all integer compositions m of n > 0. This formula is easily seen to hold by identifying standard immaculate tableaux of size n with set partitions of { 1, 2, ..., n }. For example, if we sum over integer compositions of 4 lexicographically, we see that 1+1+2+1+3+3+3+1 = 15 = A000110(4). - John M. Campbell, Jul 17 2017
a(n) is also the number of independent vertex sets (and vertex covers) in the (n-1)-triangular honeycomb bishop graph. - Eric W. Weisstein, Aug 10 2017
Even-numbered entries represent the numbers of configurations of identity and non-identity for alleles of a gene in n diploid individuals with distinguishable maternal and paternal alleles. - Noah A Rosenberg, Jan 28 2019
Number of partial equivalence relations (PERs) on a set with n elements (offset=1), i.e., number of symmetric, transitive (not necessarily reflexive) relations. The idea is to add a dummy element D to the set, and then take equivalence relations on the result; anything equivalent to D is then removed for the partial equivalence relation. - David Spivak, Feb 06 2019
Number of words of length n+1 with no repeated letters, when letters are unlabeled. - Thomas Anton, Mar 14 2019
Named by Becker and Riordan (1948) after the Scottish-American mathematician and writer Eric Temple Bell (1883 - 1960). - Amiram Eldar, Dec 04 2020
Also the number of partitions of {1,2,...,n+1} with at most one n+1 singleton. E.g., a(3)=5: {13|24, 12|34, 123|4, 14|23, 1234}. - Yuchun Ji, Dec 21 2020
a(n) is the number of sigma algebras on the set of n elements. Note that each sigma algebra is generated by a partition of the set. For example, the sigma algebra generated by the partition {{1}, {2}, {3,4}} is {{}, {1}, {2}, {1,2}, {3,4}, {1,3,4}, {2,3,4}, {1,2,3,4}}. - Jianing Song, Apr 01 2021
a(n) is the number of P_3-free graphs on n labeled nodes. - M. Eren Kesim, Jun 04 2021
a(n) is the number of functions X:([n] choose 2) -> {+,-} such that for any ordered 3-tuple abc we have X(ab)X(ac)X(bc) not in {+-+,++-,-++}. - Robert Lauff, Dec 09 2022
From Manfred Boergens, Mar 11 2025: (Start)
The partitions in the definition can be described as disjoint covers of the set. "Covers" in general give rise to the following amendments:
For disjoint covers which may include one empty set see A186021.
For arbitrary (including non-disjoint) covers see A003465.
For arbitrary (including non-disjoint) covers which may include one empty set see A000371. (End)

			G.f. = 1 + x + 2*x^2 + 5*x^3 + 15*x^4 + 52*x^5 + 203*x^6 + 877*x^7 + 4140*x^8 + ...
From Neven Juric, Oct 19 2009: (Start)
The a(4)=15 rhyme schemes for n=4 are
  aaaa, aaab, aaba, aabb, aabc, abaa, abab, abac, abba, abbb, abbc, abca, abcb, abcc, abcd
The a(5)=52 rhyme schemes for n=5 are
  aaaaa, aaaab, aaaba, aaabb, aaabc, aabaa, aabab, aabac, aabba, aabbb, aabbc, aabca, aabcb, aabcc, aabcd, abaaa, abaab, abaac, ababa, ababb, ababc, abaca, abacb, abacc, abacd, abbaa, abbab, abbac, abbba, abbbb, abbbc, abbca, abbcb, abbcc, abbcd, abcaa, abcab, abcac, abcad, abcba, abcbb, abcbc, abcbd, abcca, abccb, abccc, abccd, abcda, abcdb, abcdc, abcdd, abcde
(End)
From _Joerg Arndt_, Apr 30 2011: (Start)
Restricted growth strings (RGS):
For n=0 there is one empty string;
for n=1 there is one string [0];
for n=2 there are 2 strings [00], [01];
for n=3 there are a(3)=5 strings [000], [001], [010], [011], and [012];
for n=4 there are a(4)=15 strings
1: [0000], 2: [0001], 3: [0010], 4: [0011], 5: [0012], 6: [0100], 7: [0101], 8: [0102], 9: [0110], 10: [0111], 11: [0112], 12: [0120], 13: [0121], 14: [0122], 15: [0123].
These are one-to-one with the rhyme schemes (identify a=0, b=1, c=2, etc.).
(End)
Consider the set S = {1, 2, 3, 4}. The a(4) = 1 + 3 + 6 + 4 + 1 = 15 partitions are: P1 = {{1}, {2}, {3}, {4}}; P21 .. P23 = {{a,4}, S\{a,4}} with a = 1, 2, 3; P24 .. P29 = {{a}, {b}, S\{a,b}} with 1 <= a < b <= 4;  P31 .. P34 = {S\{a}, {a}} with a = 1 .. 4; P4 = {S}. See the Bottomley link for a graphical illustration. - _M. F. Hasler_, Oct 26 2017

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Index entries for "core" sequences
Index entries for sequences related to juggling
Index entries for sequences related to partitions
Index entries for sequences related to rooted trees
Index entries for sequences related to Benford's law

Equals row sums of triangle A008277 (Stirling subset numbers).
Partial sums give A005001. a(n) = A123158(n, 0).
See A061462 for powers of 2 dividing a(n).
Rightmost diagonal of triangle A121207. A144293 gives largest prime factor.
Cf. A000045, A000108, A000166, A000204, A000255, A000311, A000296, A003422, A024716, A029761, A049020, A058692, A060719, A084423, A087650, A094262, A103293, A165194, A165196, A173110, A227840, A182386.
Equals row sums of triangle A152432.
Row sums, right and left borders of A212431.
A diagonal of A011971. - N. J. A. Sloane, Jul 31 2012
Diagonal of A102661. - Manfred Boergens, Mar 11 2025
Cf. A054767 (period of this sequence mod n).
Row sums are A048993. - Wolfdieter Lang, Oct 16 2014
Sequences in the Erné (1974) paper: A000110, A000798, A001035, A001927, A001929, A006056, A006057, A006058, A006059.
Bell polynomials B(n,x): A001861 (x=2), A027710 (x=3), A078944 (x=4), A144180 (x=5), A144223 (x=6), A144263 (x=7), A221159 (x=8).
Cf. A243991 (sum of reciprocals), A085686 (inv. Euler Transf.).
Cf. A000371, A003465, A186021.

type N = Integer
n_partitioned_k :: N -> N -> N
1 `n_partitioned_k` 1 = 1
1 `n_partitioned_k` _ = 0
n `n_partitioned_k` k = k * (pred n `n_partitioned_k` k) + (pred n `n_partitioned_k` pred k)
n_partitioned :: N -> N
n_partitioned 0 = 1
n_partitioned n = sum $ map (\k -> n `n_partitioned_k` k) $ [1 .. n]
-- Felix Denis, Oct 16 2012

a000110 = sum . a048993_row -- Reinhard Zumkeller, Jun 30 2013

function a(n)
    t = [zeros(BigInt, n+1) for _ in 1:n+1]
    t[1][1] = 1
    for i in 2:n+1
        t[i][1] = t[i-1][i-1]
        for j in 2:i
            t[i][j] = t[i-1][j-1] + t[i][j-1]
        end
    end
    return [t[i][1] for i in 1:n+1]
end
print(a(28)) # Paul Muljadi, May 07 2024

[Bell(n): n in [0..40]]; // Vincenzo Librandi, Feb 07 2011

A000110 := proc(n) option remember; if n <= 1 then 1 else add( binomial(n-1,i)*A000110(n-1-i),i=0..n-1); fi; end: # version 1
A := series(exp(exp(x)-1),x,60): A000110 := n->n!*coeff(A,x,n): # version 2
A000110:= n-> add(Stirling2(n, k), k=0..n): seq(A000110(n), n=0..22); # version 3, from Zerinvary Lajos, Jun 28 2007
A000110 := n -> combinat[bell](n): # version 4, from Peter Luschny, Mar 30 2011
spec:= [S, {S=Set(U, card >= 1), U=Set(Z, card >= 1)}, labeled]: G:={P=Set(Set(Atom, card>0))}: combstruct[gfsolve](G, unlabeled, x): seq(combstruct[count]([P, G, labeled], size=i), i=0..22);  # version 5, Zerinvary Lajos, Dec 16 2007
BellList := proc(m) local A, P, n; A := [1, 1]; P := [1]; for n from 1 to m - 2 do
P := ListTools:-PartialSums([A[-1], op(P)]); A := [op(A), P[-1]] od; A end: BellList(29); # Peter Luschny, Mar 24 2022

f[n_] := Sum[ StirlingS2[n, k], {k, 0, n}]; Table[ f[n], {n, 0, 40}] (* Robert G. Wilson v *)
Table[BellB[n], {n, 0, 40}] (* Harvey P. Dale, Mar 01 2011 *)
B[0] = 1; B[n_] := 1/E Sum[k^(n - 1)/(k-1)!, {k, 1, Infinity}] (* Dimitri Papadopoulos, Mar 10 2015, edited by M. F. Hasler, Nov 30 2018 *)
BellB[Range[0,40]] (* Eric W. Weisstein, Aug 10 2017 *)
b[1] = 1; k = 1; Flatten[{1, Table[Do[j = k; k += b[m]; b[m] = j;, {m, 1, n-1}]; b[n] = k, {n, 1, 40}]}] (* Vaclav Kotesovec, Sep 07 2019 *)
Table[j! Coefficient[Series[Exp[Exp[x] - 1], {x, 0, 20}], x, j], {j, 0, 20}] (* Nikolaos Pantelidis, Feb 01 2023 *)
Table[(D[Exp[Exp[x]], {x, n}] /. x -> 0)/E, {n, 0, 20}] (* Joan Ludevid, Nov 05 2024 *)

makelist(belln(n),n,0,40); /* Emanuele Munarini, Jul 04 2011 */

{a(n) = my(m); if( n<0, 0, m = contfracpnqn( matrix(2, n\2, i, k, if( i==1, -k*x^2, 1 - (k+1)*x))); polcoeff(1 / (1 - x + m[2,1] / m[1,1]) + x * O(x^n), n))}; /* Michael Somos */

{a(n) = polcoeff( sum( k=0, n, prod( i=1, k, x / (1 - i*x)), x^n * O(x)), n)}; /* Michael Somos, Aug 22 2004 */

a(n)=round(exp(-1)*suminf(k=0,1.0*k^n/k!)) \\ Gottfried Helms, Mar 30 2007 - WARNING! For illustration only: Gives silently a wrong result for n = 42 and an error for n > 42, with standard precision of 38 digits. - M. F. Hasler, Nov 30 2018

{a(n) = if( n<0, 0, n! * polcoeff( exp( exp( x + x * O(x^n)) - 1), n))}; /* Michael Somos, Jun 28 2009 */

Vec(serlaplace(exp(exp('x+O('x^66))-1))) \\ Joerg Arndt, May 26 2012

A000110(n)=sum(k=0,n,stirling(n,k,2)) \\ M. F. Hasler, Nov 30 2018

use bignum;sub a{my($n)=@;my@t=map{[(0)x($n+1)]}0..$n;$t[0][0]=1;for my$i(1..$n){$t[$i][0]=$t[$i-1][$i-1];for my$j(1..$i){$t[$i][$j]=$t[$i-1][$j-1]+$t[$i][$j-1]}}return map{$t[$][0]}0..$n-1}print join(", ",a(28)),"\n" # Paul Muljadi, May 08 2024

# The objective of this implementation is efficiency.
# m -> [a(0), a(1), ..., a(m)] for m > 0.
def A000110_list(m):
    A = [0 for i in range(m)]
    A[0] = 1
    R = [1, 1]
    for n in range(1, m):
        A[n] = A[0]
        for k in range(n, 0, -1):
            A[k-1] += A[k]
        R.append(A[0])
    return R
A000110_list(40) # Peter Luschny, Jan 18 2011

# requires python 3.2 or higher. Otherwise use def'n of accumulate in python docs.
from itertools import accumulate
A000110, blist, b = [1,1], [1], 1
for _ in range(20):
    blist = list(accumulate([b]+blist))
    b = blist[-1]
    A000110.append(b) # Chai Wah Wu, Sep 02 2014, updated Chai Wah Wu, Sep 19 2014

from sympy import bell
print([bell(n) for n in range(27)]) # Michael S. Branicky, Dec 15 2021

from functools import cache
@cache
def a(n, k=0): return int(n < 1) or k*a(n-1, k) + a(n-1, k+1)
print([a(n) for n in range(27)])  # Peter Luschny, Jun 14 2022

from sage.combinat.expnums import expnums2; expnums2(30, 1) # Zerinvary Lajos, Jun 26 2008

[bell_number(n) for n in (0..40)] # G. C. Greubel, Jun 13 2019

E.g.f.: exp(exp(x) - 1).
Recurrence: a(n+1) = Sum_{k=0..n} a(k)*binomial(n, k).
a(n) = Sum_{k=0..n} Stirling2(n, k).
a(n) = Sum_{j=0..n-1} (1/(n-1)!)*A000166(j)*binomial(n-1, j)*(n-j)^(n-1). - André F. Labossière, Dec 01 2004
G.f.: (Sum_{k>=0} 1/((1-k*x)*k!))/exp(1) = hypergeom([-1/x], [(x-1)/x], 1)/exp(1) = ((1-2*x)+LaguerreL(1/x, (x-1)/x, 1)+x*LaguerreL(1/x, (2*x-1)/x, 1))*Pi/(x^2*sin(Pi*(2*x-1)/x)), where LaguerreL(mu, nu, z) = (gamma(mu+nu+1)/(gamma(mu+1)*gamma(nu+1)))* hypergeom([-mu], [nu+1], z) is the Laguerre function, the analytic extension of the Laguerre polynomials, for mu not equal to a nonnegative integer. This generating function has an infinite number of poles accumulating in the neighborhood of x=0. - Karol A. Penson, Mar 25 2002
a(n) = exp(-1)*Sum_{k >= 0} k^n/k! [Dobinski]. - Benoit Cloitre, May 19 2002
a(n) is asymptotic to n!*(2 Pi r^2 exp(r))^(-1/2) exp(exp(r)-1) / r^n, where r is the positive root of r exp(r) = n. See, e.g., the Odlyzko reference.
a(n) is asymptotic to b^n*exp(b-n-1/2)*sqrt(b/(b+n)) where b satisfies b*log(b) = n - 1/2 (see Graham, Knuth and Patashnik, Concrete Mathematics, 2nd ed., p. 493). - Benoit Cloitre, Oct 23 2002, corrected by Vaclav Kotesovec, Jan 06 2013
Lovasz (Combinatorial Problems and Exercises, North-Holland, 1993, Section 1.14, Problem 9) gives another asymptotic formula, quoted by Mezo and Baricz. - N. J. A. Sloane, Mar 26 2015
G.f.: Sum_{k>=0} x^k/(Product_{j=1..k} (1-j*x)) (see Klazar for a proof). - Ralf Stephan, Apr 18 2004
a(n+1) = exp(-1)*Sum_{k>=0} (k+1)^(n)/k!. - Gerald McGarvey, Jun 03 2004
For n>0, a(n) = Aitken(n-1, n-1) [i.e., a(n-1, n-1) of Aitken's array (A011971)]. - Gerald McGarvey, Jun 26 2004
a(n) = Sum_{k=1..n} (1/k!)*(Sum_{i=1..k} (-1)^(k-i)*binomial(k, i)*i^n + 0^n). - Paul Barry, Apr 18 2005
a(n) = A032347(n) + A040027(n+1). - Jon Perry, Apr 26 2005
a(n) = (2*n!/(Pi*e))*Im( Integral_{x=0..Pi} e^(e^(e^(ix))) sin(nx) dx ) where Im denotes imaginary part [Cesaro]. - David Callan, Sep 03 2005
O.g.f.: 1/(1-x-x^2/(1-2*x-2*x^2/(1-3*x-3*x^2/(.../(1-n*x-n*x^2/(...)))))) (continued fraction due to Ph. Flajolet). - Paul D. Hanna, Jan 17 2006
From Karol A. Penson, Jan 14 2007: (Start)
Representation of Bell numbers B(n), n=1,2,..., as special values of hypergeometric function of type (n-1)F(n-1), in Maple notation: B(n)=exp(-1)*hypergeom([2,2,...,2],[1,1,...,1],1), n=1,2,..., i.e., having n-1 parameters all equal to 2 in the numerator, having n-1 parameters all equal to 1 in the denominator and the value of the argument equal to 1.
Examples:
B(1)=exp(-1)*hypergeom([],[],1)=1
B(2)=exp(-1)*hypergeom([2],[1],1)=2
B(3)=exp(-1)*hypergeom([2,2],[1,1],1)=5
B(4)=exp(-1)*hypergeom([2,2,2],[1,1,1],1)=15
B(5)=exp(-1)*hypergeom([2,2,2,2],[1,1,1,1],1)=52
(Warning: this formula is correct but its application by a computer may not yield exact results, especially with a large number of parameters.)
(End)
a(n+1) = 1 + Sum_{k=0..n-1} Sum_{i=0..k} binomial(k,i)*(2^(k-i))*a(i). - Yalcin Aktar, Feb 27 2007
a(n) = [1,0,0,...,0] T^(n-1) [1,1,1,...,1]', where T is the n X n matrix with main diagonal {1,2,3,...,n}, 1's on the diagonal immediately above and 0's elsewhere. [Meier]
a(n) = ((2*n!)/(Pi * e)) * ImaginaryPart(Integral[from 0 to Pi](e^e^e^(i*theta))*sin(n*theta) dtheta). - Jonathan Vos Post, Aug 27 2007
From Tom Copeland, Oct 10 2007: (Start)
a(n) = T(n,1) = Sum_{j=0..n} S2(n,j) = Sum_{j=0..n} E(n,j) * Lag(n,-1,j-n) = Sum_{j=0..n} [ E(n,j)/n! ] * [ n!*Lag(n,-1, j-n) ] where T(n,x) are the Bell / Touchard / exponential polynomials; S2(n,j), the Stirling numbers of the second kind; E(n,j), the Eulerian numbers; and Lag(n,x,m), the associated Laguerre polynomials of order m. Note that E(n,j)/n! = E(n,j) / (Sum_{k=0..n} E(n,k)).
The Eulerian numbers count the permutation ascents and the expression [n!*Lag(n,-1, j-n)] is A086885 with a simple combinatorial interpretation in terms of seating arrangements, giving a combinatorial interpretation to n!*a(n) = Sum_{j=0..n} E(n,j) * [n!*Lag(n,-1, j-n)].
(End)
Define f_1(x), f_2(x), ... such that f_1(x)=e^x and for n=2,3,... f_{n+1}(x) = (d/dx)(x*f_n(x)). Then for Bell numbers B_n we have B_n=1/e*f_n(1). - Milan Janjic, May 30 2008
a(n) = (n-1)! Sum_{k=1..n} a(n-k)/((n-k)! (k-1)!) where a(0)=1. - Thomas Wieder, Sep 09 2008
a(n+k) = Sum_{m=0..n} Stirling2(n,m) Sum_{r=0..k} binomial(k,r) m^r a(k-r). - David Pasino (davepasino(AT)yahoo.com), Jan 25 2009. (Umbrally, this may be written as a(n+k) = Sum_{m=0..n} Stirling2(n,m) (a+m)^k. - N. J. A. Sloane, Feb 07 2009)
Sum_{k=1..n-1} a(n)*binomial(n,k) = Sum_{j=1..n}(j+1)*Stirling2(n,j+1). - [Zhao] - R. J. Mathar, Jun 24 2024
From Thomas Wieder, Feb 25 2009: (Start)
a(n) = Sum_{k_1=0..n+1} Sum_{k_2=0..n}...Sum_{k_i=0..n-i}...Sum_{k_n=0..1}
delta(k_1,k_2,...,k_i,...,k_n)
where delta(k_1,k_2,...,k_i,...,k_n) = 0 if any k_i > k_(i+1) and k_(i+1) <> 0
and delta(k_1,k_2,...,k_i,...,k_n) = 1 otherwise.
(End)
Let A be the upper Hessenberg matrix of order n defined by: A[i,i-1]=-1, A[i,j]:=binomial(j-1,i-1), (i<=j), and A[i,j]=0 otherwise. Then, for n>=1, a(n)=det(A). - Milan Janjic, Jul 08 2010
G.f. satisfies A(x) = (x/(1-x))*A(x/(1-x)) + 1. - Vladimir Kruchinin, Nov 28 2011
G.f.: 1 / (1 - x / (1 - 1*x / (1 - x / (1 - 2*x / (1 - x / (1 - 3*x / ... )))))). - Michael Somos, May 12 2012
a(n+1) = Sum_{m=0..n} Stirling2(n, m)*(m+1), n >= 0. Compare with the third formula for a(n) above. Here Stirling2 = A048993. - Wolfdieter Lang, Feb 03 2015
G.f.: (-1)^(1/x)*((-1/x)!/e + (!(-1-1/x))/x) where z! and !z are factorial and subfactorial generalized to complex arguments. - Vladimir Reshetnikov, Apr 24 2013
The following formulas were proposed during the period Dec 2011 - Oct 2013 by Sergei N. Gladkovskii: (Start)
E.g.f.: exp(exp(x)-1) = 1 + x/(G(0)-x); G(k) = (k+1)*Bell(k) + x*Bell(k+1) - x*(k+1)*Bell(k)*Bell(k+2)/G(k+1) (continued fraction).
G.f.: W(x) = (1-1/(G(0)+1))/exp(1); G(k) = x*k^2 + (3*x-1)*k - 2 + x - (k+1)*(x*k+x-1)^2/G(k+1); (continued fraction Euler's kind, 1-step).
G.f.: W(x) = (1 + G(0)/(x^2-3*x+2))/exp(1); G(k) = 1 - (x*k+x-1)/( ((k+1)!) - (((k+1)!)^2)*(1-x-k*x+(k+1)!)/( ((k+1)!)*(1-x-k*x+(k+1)!) - (x*k+2*x-1)*(1-2*x-k*x+(k+2)!)/G(k+1))); (continued fraction).
G.f.: A(x) = 1/(1 - x/(1-x/(1 + x/G(0)))); G(k) = x - 1 + x*k + x*(x-1+x*k)/G(k+1); (continued fraction, 1-step).
G.f.: -1/U(0) where U(k) = x*k - 1 + x - x^2*(k+1)/U(k+1); (continued fraction, 1-step).
G.f.: 1 + x/U(0) where U(k) = 1 - x*(k+2) - x^2*(k+1)/U(k+1); (continued fraction, 1-step).
G.f.: 1 + 1/(U(0) - x) where U(k) = 1 + x - x*(k+1)/(1 - x/U(k+1)); (continued fraction, 2-step).
G.f.: 1 + x/(U(0)-x) where U(k) = 1 - x*(k+1)/(1 - x/U(k+1)); (continued fraction, 2-step).
G.f.: 1/G(0) where G(k) = 1 - x/(1 - x*(2*k+1)/(1 - x/(1 - x*(2*k+2)/G(k+1) ))); (continued fraction).
G.f.: G(0)/(1+x) where G(k) = 1 - 2*x*(k+1)/((2*k+1)*(2*x*k-1) - x*(2*k+1)*(2*k+3)*(2*x*k-1)/(x*(2*k+3) - 2*(k+1)*(2*x*k+x-1)/G(k+1) )); (continued fraction).
G.f.: -(1+2*x) * Sum_{k >= 0} x^(2*k)*(4*x*k^2-2*k-2*x-1) / ((2*k+1) * (2*x*k-1)) * A(k) / B(k) where A(k) = Product_{p=0..k} (2*p+1), B(k) = Product_{p=0..k} (2*p-1) * (2*x*p-x-1) * (2*x*p-2*x-1).
G.f.: (G(0) - 1)/(x-1) where G(k) = 1 - 1/(1-k*x)/(1-x/(x-1/G(k+1) )); (continued fraction).
G.f.: 1 + x*(S-1) where S = Sum_{k>=0} ( 1 + (1-x)/(1-x-x*k) )*(x/(1-x))^k/Product_{i=0..k-1} (1-x-x*i)/(1-x).
G.f.: (G(0) - 2)/(2*x-1) where G(k) = 2 - 1/(1-k*x)/(1-x/(x-1/G(k+1) )); (continued fraction).
G.f.: -G(0) where G(k) = 1 - (x*k - 2)/(x*k - 1 - x*(x*k - 1)/(x + (x*k - 2)/G(k+1) )); (continued fraction).
G.f.: G(0) where G(k) = 2 - (2*x*k - 1)/(x*k - 1 - x*(x*k - 1)/(x + (2*x*k - 1)/G(k+1) )); (continued fraction).
G.f.: (G(0) - 1)/(1+x) where G(k) = 1 + 1/(1-k*x)/(1-x/(x+1/G(k+1) )); (continued fraction).
G.f.: 1/(x*(1-x)*G(0)) - 1/x where G(k) = 1 - x/(x - 1/(1 + 1/(x*k-1)/G(k+1) )); (continued fraction).
G.f.: 1 + x/( Q(0) - x ) where Q(k) = 1 + x/( x*k - 1 )/Q(k+1); (continued fraction).
G.f.: 1+x/Q(0), where Q(k) = 1 - x - x/(1 - x*(k+1)/Q(k+1)); (continued fraction).
G.f.: 1/(1-x*Q(0)), where Q(k) = 1 + x/(1 - x + x*(k+1)/(x - 1/Q(k+1))); (continued fraction).
G.f.: Q(0)/(1-x), where Q(k) = 1 - x^2*(k+1)/( x^2*(k+1) - (1-x*(k+1))*(1-x*(k+2))/Q(k+1) ); (continued fraction).
(End)
a(n) ~ exp(exp(W(n))-n-1)*n^n/W(n)^(n+1/2), where W(x) is the Lambert W-function. - Vladimir Reshetnikov, Nov 01 2015
a(n) ~ n^n * exp(n/W(n)-1-n) / (sqrt(1+W(n)) * W(n)^n). - Vaclav Kotesovec, Nov 13 2015
From Natalia L. Skirrow, Apr 13 2025: (Start)
By taking logarithmic derivatives of the equivalent to Kotesovec's asymptotic for Bell polynomials at x=1, we obtain properties of the nth row of A008277 as a statistical distribution (where W=W(n),X=W(n)+1)
a(n+1)/a(n) ~ n/W + W/(2*(W+1)^2) is 1 more than the expectation.
(2*a(n+1)+a(n+2))/a(n) - (a(n+1)/a(n))^2 - a(n+2)/a(n+1) ~ n/(W*X)+1/(2*X^2)-3/(2*X^3)+1/X^4 is 1 more than the variance.
(This is a complete asymptotic characterisation, since they converge to normal distributions; see Harper, 1967)
(End)
a(n) are the coefficients in the asymptotic expansion of -exp(-1)*(-1)^x*x*Gamma(-x,0,-1), where Gamma(a,z0,z1) is the generalized incomplete Gamma function. - Vladimir Reshetnikov, Nov 12 2015
a(n) = 1 + floor(exp(-1) * Sum_{k=1..2*n} k^n/k!). - Vladimir Reshetnikov, Nov 13 2015
a(p^m) == m+1 (mod p) when p is prime and m >= 1 (see Lemma 3.1 in the Hurst/Schultz reference). - Seiichi Manyama, Jun 01 2016
a(n) = Sum_{k=0..n} hypergeom([1, -k], [], 1)*Stirling2(n+1, k+1) = Sum_{k=0..n} A182386(k)*Stirling2(n+1, k+1). - Mélika Tebni, Jul 02 2022
For n >= 1, a(n) = Sum_{i=0..n-1} a(i)*A074664(n-i). - Davide Rotondo, Apr 21 2024
a(n) is the n-th derivative of e^e^x divided by e at point x=0. - Joan Ludevid, Nov 05 2024

Edited by M. F. Hasler, Nov 30 2018

A036039 Irregular triangle of multinomial coefficients of integer partitions read by rows (in Abramowitz and Stegun ordering) giving the coefficients of the cycle index polynomials for the symmetric groups S_n.

Original entry on oeis.org

1, 1, 1, 2, 3, 1, 6, 8, 3, 6, 1, 24, 30, 20, 20, 15, 10, 1, 120, 144, 90, 40, 90, 120, 15, 40, 45, 15, 1, 720, 840, 504, 420, 504, 630, 280, 210, 210, 420, 105, 70, 105, 21, 1, 5040, 5760, 3360, 2688, 1260, 3360, 4032, 3360, 1260, 1120, 1344, 2520, 1120, 1680, 105, 420, 1120, 420, 112, 210, 28, 1

Offset: 1

N. J. A. Sloane

The sequence of row lengths is A000041(n), n >= 1 (partition numbers).
Number of permutations whose cycle structure is the given partition. Row sums are factorials (A000142). - Franklin T. Adams-Watters, Jan 12 2006
A relation between partition polynomials formed from these "refined" Stirling numbers of the first kind and umbral operator trees and Lagrange inversion is presented in the link "Lagrange a la Lah".
These cycle index polynomials for the symmetric group S_n are also related to a raising operator / infinitesimal generator for fractional integro-derivatives, involving the digamma function and the Riemann zeta function values at positive integers, and to the characteristic polynomial for the adjacency matrix of complete n-graphs A055137 (cf. MathOverflow link). - Tom Copeland, Nov 03 2012
In the Lang link, replace all x(n) by t to obtain A132393. Furthermore replace x(1) by t and all other x(n) by 1 to obtain A008290. See A274760. - Tom Copeland, Nov 06 2012, Oct 29 2015 - corrected by Johannes W. Meijer, Jul 28 2016
The umbral compositional inverses of these polynomials are formed by negating the indeterminates x(n) for n>1, i.e., P(n,P(.,x(1),-x(2),-x(3),...),x(2),x(3),...) = x(1)^n (cf. A130561 for an example of umbral compositional inversion). The polynomials are an Appell sequence in x(1), i.e., dP(n,x(1))/dx(1) = n P(n-1, x(1)) and (P(.,x)+y)^n=P(n,x+y) umbrally, with P(0,x(1))=1. - Tom Copeland, Nov 14 2014
Regarded as the coefficients of the partition polynomials listed by Lang, a signed version of these polynomials IF(n,b1,b2,...,bn) (n! times polynomial on page 184 of Airault and Bouali) provides an inversion of the Faber polynomials F(n,b1,b2,...,bn) (page 52 of Bouali, A263916, and A115131). For example, F(3, IF(1,b1), IF(2,b1,b2)/2!, IF(3,b1,b2,b3)/3!) = b3 and IF(3, F(1,b1), F(2,b1,b2), F(3,b1,b2,b3))/3! = b3 with F(1,b1) = -b1. (Compare with A263634.) - Tom Copeland, Oct 28 2015; Sep 09 2016
The e.g.f. for the row partition polynomials is Sum_{n>=0} P_n(b_1,...,b_n) x^n/n! = exp[Sum_{n>=1} b_n x^n/n], or, exp[P.(b_1,...,b_n)x] = exp[-], expressed umbrally with <"power series"> denoting umbral evaluation (b.)^n = b_n within the power series. This e.g.f. is central to the paper by Maxim and Schuermannn on characteristic classes (cf. Friedrich and McKay also). - Tom Copeland, Nov 11 2015
The elementary Schur polynomials are given by S(n,x(1),x(2),...,x(n)) = P(n,x(1), 2*x(2),...,n*x(n)) / n!. See p. 12 of Carrell. - Tom Copeland, Feb 06 2016
These partition polynomials are also related to the Casimir invariants associated to quantum density states on p. 3 of Boya and Dixit and pp. 5 and 6 of Byrd and Khaneja. - Tom Copeland, Jul 24 2017
With the indeterminates (x_1,x_2,x_3,...) = (t,-c_2*t,-c_3*t,...) with c_n >0, umbrally P(n,a.) = P(n,t)|{t^n = a_n} = 0 and P(j,a.)P(k,a.) = P(j,t)P(k,t)|{t^n =a_n} = d_{j,k} >= 0 is the coefficient of x^j/j!*y^k/k! in the Taylor series expansion of the formal group law FGL(x,y) = f[f^{-1}(x)+f^{-1}(y)], where a_n are the inversion partition polynomials for calculating f(x) from the coefficients of the series expansion of f^{-1}(x) given in A133932. - Tom Copeland, Feb 09 2018
For relation to the Witt symmetric functions, as well as the basic power, elementary, and complete symmetric functions, see the Borger link p. 295. For relations to diverse zeta functions, determinants, and paths on graphs, see the MathOverflow question Cycling Through the Zeta Garden. - Tom Copeland, Mar 25 2018
Chmutov et al. identify the partition polynomials of this entry with the one-part Schur polynomials and assert that any linear combination with constant coefficients of these polynomials is a tau function for the KP hierarchy. - Tom Copeland, Apr 05 2018
With the indeterminates in the partition polynomials assigned as generalized harmonic numbers, i.e., as partial sums of the Dirichlet series for the Riemann zeta function, zeta(n), for integer n > 1, sums of simple normalizations of these polynomials give either unity or simple sums of consecutive zeta(n) (cf. Hoffman). Other identities involving these polynomials can be found in the Choi reference in Hoffman's paper. - Tom Copeland, Oct 05 2019
On p. 39 of Ma Luo's thesis is the e.g.f. of rational functions r_n obtained through the (umbral) formula 1/(1-r.T) = exp[log(1+P.T)], a differently signed e.g.f. of this entry, where (P.)^n = P_n are Eisenstein elliptic functions. P. 38 gives the example of 4! * r_4 as the signed 4th row partition polynomial of this entry. This series is equated through a simple proportionality factor to the Zagier Jacobi form on p. 25. Recurrence relations for the P_n are given on p. 24 involving the normalized k-weight Eisenstein series G_k introduced on p. 23 and related to the Bernoulli numbers. - Tom Copeland, Oct 16 2019
The Chern characteristic classes or forms of complex vector bundles and the characteristic polynomials of curvature forms for a smooth manifold can be expressed in terms of this entry's partition polynomials with the associated traces, or power sum polynomials, as the indeterminates. The Chern character is the e.g.f. of these traces and so its coefficients are given by the Faber polynomials with this entry's partition polynomials as the indeterminates. See the Mathoverflow question "A canonical reference for Chern characteristic classes". - Tom Copeland, Nov 04 2019
For an application to the physics of charged fermions in an external field, see Figueroa et al. - Tom Copeland, Dec 05 2019
Konopelchenko, in Proposition 5.2, p. 19, defines an operator P_k that is a differently signed operator version of the partition polynomials of this entry divided by a factorial. These operators give rise to bilinear Hirota equations for the KP hierarchy. These partition polynomials are also presented in Hopf algebras of symmetric functions by Cartier. - Tom Copeland, Dec 18 2019
For relationship of these partition polynomials to calculations of Pontryagin classes and the Riemann xi function, see A231846. - Tom Copeland, May 27 2020
Luest and Skliros summarize on p. 298 many of the properties of the cycle index polynomials given here; and Bianchi and Firrotta, a few on p. 6. - Tom Copeland, Oct 15 2020
From Tom Copeland, Oct 15 2020: (Start)
With a_n = n! * b_n = (n-1)! * c_n for n  > 0, represent a function with f(0) = a_0 = b_0 = 1 as an
A) exponential generating function (e.g.f), or formal Taylor series: f(x) = e^{a.x} = 1 + Sum_{n > 0} a_n * x^n/n!
B) ordinary generating function (o.g.f.), or formal power series: f(x)  = 1/(1-b.x) = 1 + Sum_{n > 0}  b_n * x^n
C) logarithmic generating function (l.g.f): f(x) = 1 - log(1 - c.x) = 1 + Sum_{n > 0}  c_n * x^n /n.
Expansions of log(f(x)) are given in
I) A127671 and A263634 for the e.g.f: log[ e^{a.*x} ] =  e^{L.(a_1,a_2,...)x} = Sum_{n > 0} L_n(a_1,...,a_n) * x^n/n!, the logarithmic polynomials, cumulant expansion polynomials
II) A263916 for the o.g.f.: log[ 1/(1-b.x) ] =  log[ 1 - F.(b_1,b_2,...)x ] = -Sum_{n > 0} F_n(b_1,...,b_n) * x^n/n, the Faber polynomials.
Expansions of exp(f(x)-1) are given in
III) A036040 for an e.g.f: exp[ e^{a.x} - 1 ] = e^{BELL.(a_1,...)x}, the Bell/Touchard/exponential partition polynomials, a.k.a. the Stirling partition polynomials of the second kind
IV) A130561 for an o.g.f.: exp[ b.x/(1-b.x) ] = e^{LAH.(b.,...)x}, the Lah partition polynomials
V) A036039 for an l.g.f.: exp[ -log(1-c.x) ] =  e^{CIP.(c_1,...)x}, the cycle index polynomials of the symmetric groups S_n, a.k.a. the Stirling partition polynomials of the first kind.
Since exp and log are a compositional inverse pair, one can extract the indeterminates of the log set of partition polynomials from the exp set and vice versa. For a discussion of the relations among these polynomials and the combinatorics of connected and disconnected graphs/maps, see Novak and LaCroix on classical moments and cumulants and the two books on statistical mechanics referenced in A036040. (End)

			The partition array T(n, k) begins (see the W. Lang link for rows 1..10):
  n\k   1    2    3    4    5    6    7    8    9   10   11  12   13  14 15 ...
  1:    1
  2:    1    1
  3:    2    3    1
  4:    6    8    3    6    1
  5:   24   30   20   20   15   10    1
  6:  120  144   90   40   90  120   15   40   45   15    1
  7:  720  840  504  420  504  630  280  210  210  420  105  70  105  21  1
... reformatted by _Wolfdieter Lang_, May 25 2019

Abramowitz and Stegun, Handbook, p. 831, column labeled "M_2".

David W. Wilson, Table of n, a(n) for n = 1..11731 (rows 1 through 26).
Milton Abramowitz and Irene A. Stegun, editors, Multinomials: M_1, M_2 and M_3, Handbook of Mathematical Functions, December 1972, pp. 831-2.
H. Airault and A. Bouali, Differential calculus on the Faber polynomials, Bulletin des Sciences Mathématiques, Volume 130, Issue 3, April-May 2006, Pages 179-222.
M. Bianchi and M. Firrotta, DDF operators, open string coherent states and their scattering amplitudes, arXiv:1902.07016 [hep-th], 2019.
J. Borger, Witt vectors, semirings, and total positivity, arXiv:1310.3013 [math.CO], 2015.
A. Bouali, Faber polynomials Cayley-Hamilton equation and Newton symmetric functions, Bulletin des Sciences Mathématiques, Volume 130, Issue 1, Jan-Feb 2006, Pages 49-70.
L. Boya and K. Dixit, Geometry of density states, arXiv:808.1930 [quant-phy], 2017.
M. Byrd and N. Khaneja, Characterization of the positivity of the density matrix in terms of the coherence vector representation , arXiv:quant-phy/0302024, 2003.
S. Carrell, Combinatorics of the KP Hierarchy, Thesis, University of Waterloo, Ontario, Canada, 2009.
P. Cartier, A primer of Hopf algebras, preprint, Institut des Hautes Etudes Scientifiques, France, 2006, pp. 56 and 57.
S. Chmutov, M. Kazarian, and S. Lando, Polynomial graph invariants and the KP hierarchy , arXiv:1803.09800 [math.CO], p. 16-17, 2018.
Tom Copeland, Lagrange a la Lah, 2011; Riemann zeta function at positive integers and an Appell sequence of polynomials, 2012; The creation / raising operators for Appell sequences, 2015; Generators, Inversion, and Matrix, Binomial, and Integral Transforms, 2015; Addendum to Elliptic Lie Triad, 2015; Formal group laws and binomial Sheffer polynomials, 2018; Appells and Roses: Newton, Leibniz, Euler, Riemann and Symmetric Polynomials, 2020.
Mark Dominus, Cycle classes of permutations [From _Wouter Meeussen_, Jun 26 2009]
G. Duchamp, Important formulas in combinatorics: The exponential formula, a Mathoverflow answer, 2015.
H. Figueroa and J. Gracia-Bondia, Combinatorial Hopf algebras in quantum field theory I, arXiv:0408145 [hep-th], 2005, (pp. 40-41).
R. Friedrich and J. McKay, Formal groups, Witt vectors and free probability, arXiv:1204.6522 [math.OA], 2012.
M. Hoffman, Harmonic-number summation identities, symmetric functions, and multiple zeta values, arXiv:1602.03198 [math.NT], p. 2-3, 2016.
B. Konopelchenko, Quantum deformations of associative algebras and integrable systems, arXiv:0802.3022 [nlin.SI], 2008.
Wolfdieter Lang, First ten rows and polynomials.
D. Luest and D. Skliros, Handle Operators in String Theory, arXiv:1912.01055 [hep-th], 2019.
Ma Luo, Algebraic De Rham Theory for Completions of Fundamental Groups of Moduli Spaces of Elliptic Curves, Ph.D. dissertation, Math Dept., Duke Univ., 2018.
Ma Luo, Algebraic De Rham Theory for Completions of Fundamental Groups of Moduli Spaces of Elliptic Curves, arXiv:1804.06977 [math.NT], 2019.
MathOverflow, Cycling through the Zeta Garden: Zeta functions for graphs, cycle index polynomials, and determinants, a question posed by Tom Copeland, 2012.
MathOverflow, Canonical reference for Chern characteristic classes, a question posed by Tom Copeland, 2019.
L. Maxim and J. Schuermann, Equivariant characteristic classes of external and symmetric products of varieties, arXiv:1508.04356 [math.AG], 2015.
R. Szabo, N=2 gauge theories, instanton moduli spaces and geometric representation theory, arXiv:1507.00685 [hep-th], 2015.
Andrei Vieru, Analytic renormalization of multiple zeta functions. Geometry and combinatorics of generalized Euler reflection formula for MZV, arXiv preprint arXiv:1601.04703 [math.NT], 2016.

Cf. A036036, A036037, A036038, A036040.
Cf. other versions based on different partition orderings: A102189 (rows reversed), A181897, A319192.
Cf. A094587, A115131, A130561, A263634, A263916.
Cf. A133932.
Cf. A231846.
Cf. A127671.

nmax:=7: with(combinat): for n from 1 to nmax do P(n):=sort(partition(n)): for r from 1 to numbpart(n) do B(r):=P(n)[r] od: for m from 1 to numbpart(n) do s:=0: j:=0: while sA036039(n, m) := n!/ (mul((t)^q(t)*q(t)!, t=1..n)); od: od: seq(seq(A036039(n, m), m=1..numbpart(n)), n=1..nmax); # Johannes W. Meijer, Jul 14 2016
# 2nd program:
A036039 := proc(n,k)
    local a,prts,e,ai ;
    a := n! ;
    # ASPrts is implemented in A119441
    prts := ASPrts(n)[k] ;
    ai := 1;
    for e from 1 to nops(prts) do
        if e>1 then
            if op(e,prts) = op(e-1,prts) then
                ai := ai+1 ;
            else
                ai := 1;
            end if;
        end if;
        a := a/(op(e,prts)*ai) ;
    end do:
    a ;
end proc:
seq(seq(A036039(n,k),k=1..combinat[numbpart](n)),n=1..15) ; # R. J. Mathar, Dec 18 2016

aspartitions[n_]:=Reverse/@Sort[Sort/@IntegerPartitions[n]];(* Abramowitz & Stegun ordering *);
ascycleclasses[n_Integer]:=n!/(Times@@ #)&/@((#!
Range[n]^#)&/@Function[par,Count[par,# ]&/@Range[n]]/@aspartitions[n])
(* The function "ascycleclasses" is then identical with A&S multinomial M2. *)
Table[ascycleclasses[n], {n, 1, 8}] // Flatten
(* Wouter Meeussen, Jun 26 2009, Jun 27 2009 *)

def PartAS(n):
    P = []
    for k in (1..n):
        Q = [p.to_list() for p in Partitions(n, length=k)]
        for q in Q: q.reverse()
        P = P + sorted(Q)
    return P
def A036039_row(n):
    fn, C = factorial(n), []
    for q in PartAS(n):
        q.reverse()
        p = Partition(q)
        fp = 1; pf = 1
        for a, c in p.to_exp_dict().items():
            fp *= factorial(c)
            pf *= factorial(a)**c
        co = fn//(fp*pf)
        C.append(co*prod([factorial(i-1) for i in p]))
    return C
for n in (1..10):
    print(A036039_row(n)) # Peter Luschny, Dec 18 2016

T(n,k) = n!/Product_{j=1..n} j^a(n,k,j)*a(n,k,j)!, with the k-th partition of  n >= 1 in Abromowitz-Stegun order written as Product_{j=1..n} j^a(n,k,j) with nonnegative integers a(n,k,j) satisfying Sum_{j=1..n} j*a(n,k,j) = n, and the number of parts is Sum_{j=1..n} a(n,k,j) =: m(n,k). - Wolfdieter Lang, May 25 2019
Raising and lowering operators are given for the partition polynomials formed from this sequence in the link in "Lagrange a la Lah Part I" on p. 23. - Tom Copeland, Sep 18 2011
From Szabo p. 34, with b_n = q^n / (1-q^n)^2, the partition polynomials give an expansion of the MacMahon function M(q) = Product_{n>=1} 1/(1-q^n)^n = Sum_{n>=0} PL(n) q^n, the generating function for PL(n) = n! P_n(b_1,...,b_n), the number of plane partitions with sum n. - Tom Copeland, Nov 11 2015
From Tom Copeland, Nov 18 2015: (Start)
The partition polynomials of A036040 are obtained by substituting x[n]/(n-1)! for x[n] in the partition polynomials of this entry.
CIP_n(t-F(1,b1),-F(2,b1,b2),...,-F(n,b1,...,bn)) = P_n(b1,...,bn;t), where CIP_n are the partition polynomials of this entry; F(n,...), those of A263916; and P_n, those defined in my formula in A094587, e.g., P_2(b1,b2;t) = 2 b2 + 2 b1 t + t^2.
CIP_n(-F(1,b1),-F(2,b1,b2),...,-F(n,b1,...,bn)) = n! bn. (End)
From the relation to the elementary Schur polynomials given in A130561 and above, the partition polynomials of this array satisfy (d/d(x_m)) P(n,x_1,...,x_n) = (1/m) * (n!/(n-m)!) * P(n-m,x_1,...,x_(n-m)) with P(k,...) = 0 for k<0. - Tom Copeland, Sep 07 2016
Regarded as Appell polynomials in the indeterminate x(1)=u, the partition polynomials of this entry P_n(u) obey d/du P_n(u) = n * P_{n-1}(u), so the abscissas for the zeros of P_n(u) are the same as those of the extrema of P{n+1}(u). In addition, the coefficient of u^{n-1} in P_{n}(u) is zero since these polynomials are related to the characteristic polynomials of matrices with null main diagonals, and, therefore, the trace is zero, further implying the abscissa for any zero is the negative of the sum of the abscissas of the remaining zeros. This assumes all zeros are distinct and real. - Tom Copeland, Nov 10 2019

More terms from David W. Wilson

Title expanded by Tom Copeland, Oct 15 2020

cp's OEIS Frontend

A181512 Irregular triangle a(n,k) = A181415(n,k) / A036040(n,k).

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A179236 Irregular triangle T(n,k) = A096162(n,k)* A036040(n,k)* A048996(n,k)*A098546(n,k)* A178886(n,k) read by rows, 1<=k<=A000041(n).

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A178882 Triangle T(n,k) = n!* A036040(n,k) read by rows, 1 <= k <= A000041(n).

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A181417 Irregular triangle T(n,k) = binomial(n-1,m-1)*m!*A036040(n,k), where m=A036043(n,k).

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A000110 Bell or exponential numbers: number of ways to partition a set of n labeled elements.

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Haskell

Haskell

Julia

Magma

Maple

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Maxima

PARI

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PARI

PARI

PARI

Perl

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A008480 Number of ordered prime factorizations of n.

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A036036 Triangle read by rows in which row n lists all the parts of all reversed partitions of n, sorted first by length and then lexicographically.

A179236 Irregular triangle T(n,k) = A096162(n,k)* A036040(n,k)* A048996(n,k)A098546(n,k) A178886(n,k) read by rows, 1<=k<=A000041(n).

A181417 Irregular triangle T(n,k) = binomial(n-1,m-1)m!A036040(n,k), where m=A036043(n,k).