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Also known as Stirling set numbers.

S(n,k) enumerates partitions of an n-set into k nonempty subsets.

The o.g.f. for the sequence of diagonal k (k=0 for the main diagonal) is G(k,x) = ((x^k)/(1-x)^(2*k+1))*Sum_{m=0..k-1} A008517(k,m+1)*x^m. A008517 is the second-order Eulerian triangle. - Wolfdieter Lang, Oct 14 2005

From Philippe Deléham, Nov 14 2007: (Start)

Sum_{k=0..n} S(n,k)*x^k = B_n(x), where B_n(x) = Bell polynomials.

The first few Bell polynomials are:

B_0(x) = 1;

B_1(x) = 0 + x;

B_2(x) = 0 + x + x^2;

B_3(x) = 0 + x + 3x^2 + x^3;

B_4(x) = 0 + x + 7x^2 + 6x^3 + x^4;

B_5(x) = 0 + x + 15x^2 + 25x^3 + 10x^4 + x^5;

B_6(x) = 0 + x + 31x^2 + 90x^3 + 65x^4 + 15x^5 + x^6;

(End)

This is the Sheffer triangle (1, exp(x) - 1), an exponential (binomial) convolution triangle. The a-sequence is given by A006232/A006233 (Cauchy sequence). The z-sequence is the zero sequence. See the link under A006232 for the definition and use of these sequences. The row sums give A000110 (Bell), and the alternating row sums give A000587 (see the Philippe Deléham formulas and crossreferences below). - Wolfdieter Lang, Oct 16 2014

Also the inverse Bell transform of the factorial numbers (A000142). For the definition of the Bell transform see A264428 and for cross-references A265604. - Peter Luschny, Dec 31 2015

From Wolfdieter Lang, Feb 21 2017: (Start)

The transposed (trans) of this lower triagonal Sheffer matrix of the associated type S = (1, exp(x) - 1) (taken as N X N matrix for arbitrarily large N) provides the transition matrix from the basis {x^n/n!}, n >= 0, to the basis {y^n/n!}, n >= 0, with y^n/n! = Sum_{m>=n} S^{trans}(n, m) x^m/m! = Sum_{m>=0} x^m/m!*S(m, n).

The Sheffer transform with S = (g, f) of a sequence {a_n} to {b_n} for n >= 0, in matrix notation vec(b) = S vec(a), satisfies, with e.g.f.s A and B, B(x) = g(x)*A(f(x)) and B(x) = A(y(x)) identically, with vec(xhat) = S^{trans,-1} vec(yhat) in symbolic notation with vec(xhat)_n = x^n/n! (similarly for vec(yhat)).

(End)

For k >= 1 S(n, k) = h^{(k)}{n-k}, the complete homogeneous symmetric function of the k symbols 1,2, ..., k, of degree n-k. Thus S(n, k) is for k >= 1 the (dimensionless) volume of the multichoose(k, n-k) = binomial(n-1, k-1) polytopes of dimension n-k with (dimensionless) side lengths from the set {1, 2, ..., k}. See an example below. - _Wolfdieter Lang, May 26 2017

Number of partitions of {1, 2, ..., n+1} into k+1 nonempty subsets such that no subset contains two adjacent numbers. - Thomas Anton, Sep 26 2022

Abscissa of the image produced after n alternating reflections of (1,1) over the x and y axes respectively. Similarly, the ordinate of the image produced after n alternating reflections of (1,1) over the y and x axes respectively. - Wesley Ivan Hurt, Jul 06 2013

Values of Bell polynomials: ways of placing n labeled balls into n unlabeled (but 2-colored) boxes.

First column of the square of the matrix exp(P)/exp(1) given in A011971. - Gottfried Helms, Mar 30 2007

Base matrix in A011971, second power in A078937, third power in A078938, fourth power in A078939. - Gottfried Helms, Apr 08 2007

Equals row sums of triangle A144061. - Gary W. Adamson, Sep 09 2008

Equals eigensequence of triangle A109128. - Gary W. Adamson, Apr 17 2009

Hankel transform is A108400. - Paul Barry, Apr 29 2009

The number of ways of putting n labeled balls into a set of bags and then putting the bags into 2 labeled boxes. An example is given below. - Peter Bala, Mar 23 2013

The f-vectors of n-dimensional hypercube are given by A038207 = exp[M*B(.,2)] = exp[M*A001861(.)] where M = A238385-I and (B(.,x))^n = B(n,x) are the Bell polynomials (cf. A008277). - Tom Copeland, Apr 17 2014

Moments of the Poisson distribution with mean 2. - Vladimir Reshetnikov, May 17 2016

Exponential self-convolution of Bell numbers (A000110). - Vladimir Reshetnikov, Oct 06 2016

The list partition transform of a sequence a(n) for which a(0)=1 is illustrated by:

b_0 = 1

b_1 = -a_1

b_2 = -a_2 + 2 a_1^2

b_3 = -a_3 + 6 a_2 a_1 - 6 a_1^3

b_4 = -a_4 + 8 a_3 a_1 + 6 a_2^2 - 36 a_2 a_1^2 + 24 a_1^4

... .

The unsigned coefficients are A049019 with a leading 1. The sign is dependent on the partition as evident from inspection (replace a_n's by -1).

Expressed umbrally, i.e., with the umbral operation (a.)^n := a_n,

exp(a.x) exp(b.x) = exp[(a.+b.)x] = 1; i.e., (a.+b.)^n = 1 for n=0 and 0 for all other values of n.

Expressed recursively,

b_0 = 1, b_n = -Sum_{j=1..n} binomial(n,j) a_j b_{n-j}; which is conditionally self-inverse, i.e., the roles of a_k and b_k may be reversed with a_0 = b_0 = 1.

Expressed in matrix form, b_n form the first column of B = matrix inverse of A .

A = Pascal matrix diagonally multiplied by a_n, i.e., A_{n,k} = binomial(n,k)* a_{n-k}.

Some examples of reciprocal pairs of sequences under these operations are:

1) A084358 and -A000262 with the first term set to 1.

2) (1,-1,0,0,...) and (0!,1!,2!,3!,...) with the unsigned associated matrices A128229 and A094587.

3) (1,-1,-1,-1,...) and A000670.

4) A000110 and A000587.

5) (1,-2,-2,0,0,0,...) and (0! c_1,1! c_2,2! c_3,3! c_4,...) where c_n = A000129(n) with the associated matrices A110327 and A110330.

6) (1,-2,2,0,0,0,...) and (1!,2!,3!,4!,...).

7) Sequences of rising and signed lowering factorials form reciprocal pairs where a_n = (-1)^n m!/(m-n)! and b_n = (m-1+n)!/(m-1)! for m=0,1,2,... .

Denote the action of the list partition transform on the sequence a. or an invertible matrix M by LPT(a.) = b. or LPT(M)= M^(-1).

If the matrix equation M = exp(T) also holds, then exp[a.*T]*exp[b.*T] = exp[(a.+b.)*T] = I, the identity matrix, because (a.+b.)^n = delta_n, the Kronecker delta with delta_n = 1 and delta_n = 0 otherwise, i.e., (0)^n = delta_n.

Therefore, [exp(a.*T)]^(-1) = exp[b.*T] = exp[LPT(a.)*T] = LPT[exp(a.*T)].

The fundamental Pascal (A007318), unsigned Lah (A105278) and associated Laguerre matrices can be generated by exponentiation of special infinitesimal matrices (see A132440, A132710 and A132681) such that finding LPT(a.) amounts to multiplying the k'th diagonal of the fundamental matrices by a_k for every diagonal followed by matrix inversion and then extraction of the b_n factors from the first column (simplest for the Pascal formulas above).

Conversely, the inverses of matrices formed by diagonally multiplying the three fundamental matrices by a_k are given by diagonally multiplying the fundamental matrices by b_k.

If LPT(M) is defined differently as application of the top formula to a_n = M^n, then b_n = (-M)^n and the formalism could even be applied to more general sequences of matrices M., providing the reciprocal of exp[t*M.].

The group of fundamental lower triangular matrices M = exp(T) such that LPT[exp(a.*T)] = exp[LPT(a.)*T] = [exp[a.*T]]^(-1) are obtained by infinitesimal generator matrices of the form T =

0;

t(0), 0;

0, t(1), 0;

0, 0, t(2), 0;

0, 0, 0, t(3), 0;

... .

T^m has trivially vanishing terms except along the m'th subdiagonal, which is a sequence of generalized factorials:

[ t(0)*t(1)...t(m-2)*t(m-1), t(1)*t(2)...t(m-1)*t(m), t(2)*t(3)...t(m)*t(m+1), ... ].

Therefore the principal submatrices of T (given by setting t(j) = 0 for j > n-1) are nilpotent with at least [Tsub_n]^(n+1) = 0.

The general group of matrices GM[a.] = exp[a.*T] can also be obtained through diagonal multiplication of M = exp(T) by the sequence a_n, as in the Pascal matrix example above and their inverses by diagonal multiplication by b. = LPT(a.).

Weighted-mappings interpretation for the top partition equation:

Given n pre-nodes (Pre) and k post-nodes (Post), each Pre is connected to only one Post and each Post has at least one Pre connected to it (surjections or onto functions/maps). Weight each Post by -a_m where m is the number of connections to the Post.

Weight each map by the product of the Post weights and multiply by the number of maps that share the same connectivity. Sum over the possible mappings for n Pre. The result is b_n.

E.g., b_3 = [ 3 Pre to 1 Post ] + [ 3 Pre to 2 Post ] + [ 3 Pre to 3 Post ]

= [1 map with 1 Post with 3 connections] + [ 6 maps with 1 Post with 2 connections and 1 Post with 1 connection] + [6 maps with 3 Post with 1 connection each]

= -a_3 + 6 * [-a_2*(-a_1)] + 6 * [-a_1*(-a_1)*(-a_1)].

See A263633 for the complementary formulation for the reciprocal of o.g.f.s rather than e.g.f.s and computations of these partition polynomials as Gram determinants. - Tom Copeland, Dec 04 2016

The coefficients of the partition polynomials enumerate the faces of the convex, bounded polytopes called permutohedra, and the absolute value of the sum of the coefficients gives the Euler characteristic of unity for each polytope; i.e., the absolute value of the sum of each row of the array is unity. In addition, the signs of the faces alternate with dimension, and the coefficients of faces with the same dimension for each polytope have the same sign. - Tom Copeland, Nov 13 2019

With the fundamental matrix chosen to be the lower triangular Pascal matrix M, the matrix MA whose n-th diagonals are multiplied by a_n (i.e., MA_{i,j} = PM_{i,j} * a_{i-j}) gives a matrix representation of the e.g.f. associated to the Appell polynomial sequence defined by e^{a.t}e^{xt}= e^{(a.+x)t} = e^{A.(x)t} where umbrally (A.(x))^n = A_n(x) = (a. + x)^n = sum_{k=0..n} binomial(n,k) a_k x^{n-k} are the associated Appell polynomials. Left multiplication of the column vector (1,x,x^2,..) by MA gives the Appell polynomial sequence, and multiplication of the two e.g.f.s e^{a.t} and e^{b.t} corresponds to multiplication of their respective matrix representations MA and MB. Forming the reciprocal of an e.g.f. corresponds to taking the matrix inverse of its matrix representation as noted above. A263634 gives an associated modified Pascal matrix representation of the raising operator for the Appell sequence. - Tom Copeland, Nov 13 2019

The diagonal of MA consists of all ones. Let MAN be the truncated square submatrix of MA containing the coefficients of the first N Appell polynomials A_k=(a.+x)^k = Sum(j=0 to k) MAN(k,j) x^j. Then by the Cayley-Hamilton theorem (I-MAN)^N = 0; therefore, MAN^(-1) = Sum(k=1 to N) binomial(N,k) (-MAN)^{k-1} = MBN, the inverse of MAN, containing the coefficients of the first N rows of the Appell polynomials B_k(x) = (b. + x)^k = Sum(j=0 to k) MBN(k,j) x^j, which are the umbral compositional inverses of the Appell row polynomials A_k(x) of MAN; that is, A_k(B.(x)) = x^k = B_k(A.(x)), where, e.g., (A.(x))^k = A_k(x). - Tom Copeland, May 13 2020

The use of the term 'list partition transform' resulted from one of my first uses of these partition polynomials in relating A000262 to A084358 with their simple e.g.f.s. Other appropriate names would be the permutohedra polynomials since they are refined Euler characteristics of the permutohedra or the reciprocal polynomials since they give the multiplicative inverses of e.g.f.s with a constant of 1. - Tom Copeland, Oct 09 2022

Also called the Bell triangle or the Peirce triangle.

a(n,k) is the number of equivalence relations on {0, ..., n} such that k is not equivalent to n, k+1 is not equivalent to n, ..., n-1 is not equivalent to n. - Don Knuth, Sep 21 2002 [Comment revised by Thijs van Ommen (thijsvanommen(AT)gmail.com), Jul 13 2008]

Named after the New Zealand mathematician Alexander Craig Aitken (1895-1967). - Amiram Eldar, Jun 11 2021

Equals the eigensequence of A038207, the square of Pascal's triangle. - Gary W. Adamson, Apr 10 2009

The g.f. of the second binomial transform is 1/(1-2x-x/(1-2x/(1-2x-x/(1-4x/(1-2x-x/(1-6x/(1-2x-x/(1-8x/(1-... (continued fraction). - Paul Barry, Dec 04 2009

Length-n restricted growth strings (RGS) [s(0),s(1),...,s(n-1)] where s(k)<=F(k)+2 where F(0)=0 and F(k+1)=s(k+1) if s(k+1)-s(k)=2, otherwise F(k+1)=F(k); see example and Fxtbook link. - Joerg Arndt, Apr 30 2011

It appears that the infinite set of "Shifts 1 place left under N-th order binomial transform" sequences has a production matrix of:

1, N, 0, 0, 0, ...

1, 1, N, 0, 0, ...

1, 2, 1, N, 0, ...

1, 3, 3, 1, N, ...

... (where a diagonal of (N,N,N,...) is appended to the right of Pascal's triangle). a(n) in each sequence is the upper left term of M^n such that N=1 generates A000110, then (N=2 - A004211), (N=3 - A004212), (N=4 - A004213), (N=5 - A005011), ... - Gary W. Adamson, Jul 29 2011

Number of "unlabeled" hierarchical orderings on set partitions of {1..n}, see comments on A034691. - Gus Wiseman, Mar 03 2016

From Lorenzo Sauras Altuzarra, Jun 17 2022: (Start)

Number of n-variate noncontradictory conjunctions of logical equalities of literals (modulo logical equivalence).

Equivalently, number of n-variate noncontradictory Krom formulas with palindromic truth-vector (modulo logical equivalence).

a(n) <= A109457(n). (End)

Number of partitions of an n-element set into an odd number of classes. - Peter Luschny, Apr 25 2011

Let A(0) = 1, B(0) = 0; A(n+1) = Sum_{k=0..n} binomial(n,k)*B(k), B(n+1) = Sum_{k=0..n} binomial(n,k)*A(k); entry gives B sequence (cf. A024430).

Number of partitions of an n-element set into an even number of classes.

Let A(0) = 1, B(0) = 0; A(n+1) = Sum_{k=0..n} binomial(n,k)*B(k), B(n+1) = Sum_{k=0..n} binomial(n,k)*A(k); entry gives A sequence (cf. A024429).

a(n) is the number of ways of separating n terms into pairs. - Stephen Crowley, Apr 07 2007

a(n) is the n-th moment of the standard normal distribution. - Hal M. Switkay, Nov 06 2019

a(n) is the number of fixed-point free involutions in the symmetric group of degree n. - Nick Krempel, Feb 26 2020