A178866 -id:A178866 - cp's OEIS frontend

A000296 Set partitions without singletons: number of partitions of an n-set into blocks of size > 1. Also number of cyclically spaced (or feasible) partitions.

1, 0, 1, 1, 4, 11, 41, 162, 715, 3425, 17722, 98253, 580317, 3633280, 24011157, 166888165, 1216070380, 9264071767, 73600798037, 608476008122, 5224266196935, 46499892038437, 428369924118314, 4078345814329009, 40073660040755337, 405885209254049952, 4232705122975949401

Offset: 0

N. J. A. Sloane

a(n+2) = p(n+1) where p(x) is the unique degree-n polynomial such that p(k) = A000110(k) for k = 0, 1, ..., n. - Michael Somos, Oct 07 2003
Number of complete rhyming schemes.
Whereas the Bell number B(n) (A000110(n)) is the number of terms in the polynomial that expresses the n-th moment of a probability distribution as a function of the first n cumulants, these numbers give the number of terms in the corresponding expansion of the central moment as a function of the first n cumulants. - Michael Hardy (hardy(AT)math.umn.edu), Jan 26 2005
a(n) is the number of permutations on [n] for which the left-to-right maxima coincide with the descents (entries followed by a smaller number). For example, a(4) counts 2143, 3142, 3241, 4123. - David Callan, Jul 20 2005
From Gus Wiseman, Feb 10 2019: (Start)
Also the number of stable partitions of an n-cycle, where a stable partition of a graph is a set partition of the vertex set such that no edge has both ends in the same block. A bijective proof is given in David Callan's article. For example, the a(5) = 11 stable partitions are:
  {{1},{2},{3},{4},{5}}
  {{1},{2},{3,5},{4}}
  {{1},{2,4},{3},{5}}
  {{1},{2,5},{3},{4}}
  {{1,3},{2},{4},{5}}
  {{1,4},{2},{3},{5}}
  {{1},{2,4},{3,5}}
  {{1,3},{2,4},{5}}
  {{1,3},{2,5},{4}}
  {{1,4},{2},{3,5}}
  {{1,4},{2,5},{3}}
(End)
Also number of partitions of {1, 2, ..., n-1} with singletons. E.g., a(4) = 4: {1|2|3, 12|3, 13|2, 1|23}. Also number of cyclical adjacencies partitions of {1, 2, ..., n-1}. E.g., a(4) = 4: {12|3, 13|2, 1|23, 123}. The two partitions can be mapped by a Kreweras bijection. - Yuchun Ji, Feb 22 2021
Also the k-th central moment of a Poisson random variable with mean 1. a(n) = E[(X-1)^n, X~Poisson(1)]. - Thomas Dybdahl Ahle, Dec 14 2022

			a(4) = card({{{1, 2}, {3, 4}}, {{1, 4}, {2, 3}}, {{1, 3}, {2, 4}}, {{1, 2, 3, 4}}}) = 4.

Martin Gardner in Sci. Amer. May 1977.
D. E. Knuth, The Art of Computer Programming, vol. 4A, Combinatorial Algorithms, Section 7.2.1.5 (p. 436).
G. Pólya and G. Szegő, Problems and Theorems in Analysis, Springer-Verlag, NY, 2 vols., 1972, Vol. 1, p. 228.
J. Riordan, A budget of rhyme scheme counts, pp. 455-465 of Second International Conference on Combinatorial Mathematics, New York, April 4-7, 1978. Edited by Allan Gewirtz and Louis V. Quintas. Annals New York Academy of Sciences, 319, 1979.
J. Shallit, A Triangle for the Bell numbers, in V. E. Hoggatt, Jr. and M. Bicknell-Johnson, A Collection of Manuscripts Related to the Fibonacci Sequence, 1980, pp. 69-71.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Alois P. Heinz, Table of n, a(n) for n = 0..575 (first 101 terms from T. D. Noe)
Yasemin Alp and E. Gokcen Kocer, Exponential Almost-Riordan Arrays, Results Math. (2024) Vol. 79, 173.
Joerg Arndt and N. J. A. Sloane, Counting Words that are in "Standard Order".
E. Bach, Random bisection and evolutionary walks, J. Applied Probability, v. 38, pp. 582-596, 2001.
M. Bauer and O. Golinelli, Random incidence matrices: Moments of the spectral density, arXiv:cond-mat/0007127 [cond-mat.stat-mech], 2000-2001. See Sect. 3.2; J. Stat. Phys. 103, 301-307 (2001).
H. D. Becker, Solution to problem E 461, American Math Monthly 48 (1941), 701-702.
F. R. Bernhart, Catalan, Motzkin and Riordan numbers, Discr. Math., 204 (1999) 73-112.
F. R. Bernhart, Fundamental chromatic numbers, Unpublished. (Annotated scanned copy)
F. R. Bernhart & N. J. A. Sloane, Correspondence, 1977.
J. R. Britnell and M. Wildon, Bell numbers, partition moves and the eigenvalues of the random-to-top shuffle in types A, B and D, arXiv 1507.04803 [math.CO], 2015.
David Callan, On conjugates for set partitions and integer compositions [math.CO].
Pascal Caron, Jean-Gabriel Luque, Ludovic Mignot, and Bruno Patrou, State complexity of catenation combined with a boolean operation: a unified approach, arXiv preprint arXiv:1505.03474 [cs.FL], 2015.
Robert Coquereaux and Jean-Bernard Zuber, Counting partitions by genus. II. A compendium of results, arXiv:2305.01100 [math.CO], 2023. See p. 3.
Éva Czabarka, Péter L. Erdős, Virginia Johnson, Anne Kupczok and László A. Székely, Asymptotically normal distribution of some tree families relevant for phylogenetics, and of partitions without singletons, arXiv preprint arXiv:1108.6015 [math.CO], 2011.
Gesualdo Delfino and Jacopo Viti, Potts q-color field theory and scaling random cluster model, arXiv preprint arXiv:1104.4323 [hep-th], 2011.
E. A. Enneking and J. C. Ahuja, Generalized Bell numbers, Fib. Quart., 14 (1976), 67-73.
Steven R. Finch, Moments of sums, April 23, 2004. [Cached copy, with permission of the author]
Robert C. Griffiths, P. A. Jenkins, and S. Lessard, A coalescent dual process for a Wright-Fisher diffusion with recombination and its application to haplotype partitioning, arXiv preprint arXiv:1604.04145 [q-bio.PE], 2016.
Aoife Hennessy, A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 16.
V. P. Johnson, Enumeration Results on Leaf Labeled Trees, Ph. D. Dissertation, Univ. Southern Calif., 2012.
J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.
Peter Luschny, Set partitions.
Gregorio Malajovich, Complexity of sparse polynomial solving: homotopy on toric varieties and the condition metric, arXiv preprint arXiv:1606.03410 [math.NA], 2016.
Toufik Mansour and Mark Shattuck, A recurrence related to the Bell numbers, INTEGERS 11 (2011), #A67.
T. Mansour and A. O. Munagi, Set partitions with circular successions, European Journal of Combinatorics, 42 (2014), 207-216.
I. Mezo, Periodicity of the last digits of some combinatorial sequences, J. Integer Seq. 17, Article 14.1.1, 2014.
Scott Morrison, Noah Snyder, and Dylan P. Thurston, Towards the quantum exceptional series, arXiv:2402.03637 [math.QA], 2024. See p. 39.
E. Norton, Symplectic Reflection Algebras in Positive Characteristic as Ore Extensions, arXiv preprint arXiv:1302.5411 [math.RA], 2013.
Rosa Orellana, Nancy Wallace, and Mike Zabrocki, Quasipartition and planar quasipartition algebras, Sém. Lotharingien Comb., Proc. 36th Conf. Formal Power Series Alg. Comb. (2024) Vol. 91B, Art. No. 50. See p. 7.
Aleksandar Petojević, Marjana Gorjanac Ranitović, and Milinko Mandić, New equivalents for Kurepa's hypothesis for left factorial, Univ. Novi Sad (2023).
Tilman Piesk, Table showing non-singleton partitions for n = 1..6.
R. A. Proctor, Let's Expand Rota's Twelvefold Way for Counting Partitions!, arXiv:math/0606404 [math.CO], 2006-2007.
Jocelyn Quaintance and Harris Kwong, A combinatorial interpretation of the Catalan and Bell number difference tables, Integers, 13 (2013), #A29.
D. Reidenbach and J. C. Schneider, Morphically Primitive Words. In Pierre Arnoux, Nicolas Bedaride and Julien Cassaigne, editors, Proc. 6th International Conference on Words, WORDS 2007, pages 262-272. 2007. [Different from the paper with the same name, referenced below.]
Daniel Reidenbach and Johannes C. Schneider, Morphically primitive words, (2009). See Table 1.
Daniel Reidenbach and Johannes C. Schneider, Morphically primitive words, Theoretical Computer Science, (2009), 140 (21-23), pp. 2148-2161.
J. Riordan, Cached copy of paper.
Jeffrey Shallit, A Triangle for the Bell numbers, in V. E. Hoggatt, Jr. and M. Bicknell-Johnson, A Collection of Manuscripts Related to the Fibonacci Sequence, 1980, pp. 69-71.
Index entries for related partition-counting sequences

Cf. A000110, A005493, A006505, A057814, A057837.
A diagonal of triangle in A106436.
Row sums of the triangle of associated Stirling numbers of second kind A008299. - Philippe Deléham, Feb 10 2005
Row sums of the triangle of basic multinomial coefficients A178866. - Johannes W. Meijer, Jun 21 2010
Row sums of A105794. - Peter Bala, Jan 14 2015
Row sums of A261139, main diagonal of A261137.
Column k=0 of A216963.
Column k=0 of A124323.
Cf. A000126, A001610, A066982, A169985, A240936.

[1,0] cat [ n le 1 select 1 else Bell(n)-Self(n-1) : n in [1..40]]; // Vincenzo Librandi, Jun 22 2015

spec := [ B, {B=Set(Set(Z,card>1))}, labeled ]; [seq(combstruct[count](spec, size=n), n=0..30)];
with(combinat): A000296 :=n->(-1)^n + add((-1)^(j-1)*bell(n-j),j=1..n): seq(A000295(n),n=0..30); # Emeric Deutsch, Oct 29 2006
f:=exp(exp(x)-1-x): fser:=series(f, x=0, 31): 1, seq(n!*coeff(fser, x^n), n=1..23); # Zerinvary Lajos, Nov 22 2006
G:={P=Set(Set(Atom,card>=2))}: combstruct[gfsolve](G,unlabeled,x): seq(combstruct[count]([P,G,labeled], size=i), i=0..23); # Zerinvary Lajos, Dec 16 2007
# [a(0),a(1),..,a(n)]
A000296_list := proc(n)
local A, R, i, k;
if n = 0 then return 1 fi;
A := array(0..n-1);
A[0] := 1; R := 1;
for i from 0 to n-2 do
   A[i+1] := A[0] - A[i];
   A[i] := A[0];
   for k from i by -1 to 1 do
      A[k-1] := A[k-1] + A[k] od;
   R := R,A[i+1];
od;
R,A[0]-A[i] end:
A000296_list(100);  # Peter Luschny, Apr 09 2011

nn = 25; Range[0, nn]! CoefficientList[Series[Exp[Exp[x] - 1 - x], {x, 0, nn}], x]
(* Second program: *)
a[n_] := a[n] = If[n==0, 1, Sum[Binomial[n-1, i]*a[n-i-1], {i, 1, n-1}]]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Feb 06 2016, after Vladimir Kruchinin *)
spsu[,{}]:={{}};spsu[foo,set:{i_,_}]:= Join@@Function[s,Prepend[#,s]&/@spsu[ Select[foo,Complement[#, Complement[set,s]]=={}&], Complement[set,s]]]/@Cases[foo,{i,_}];
Table[Length[spsu[Select[Subsets[Range[n]],Select[Partition[Range[n],2,1,1], Function[ed,Complement[ed,#]=={}]]=={}&],Range[n]]],{n,8}] (* Gus Wiseman, Feb 10 2019 *)
s = 1; Join[{1}, Table[s = BellB[n] - s, {n, 0, 25}]] (* Vaclav Kotesovec, Jun 20 2022 *)

a(n):=if n=0 then 1 else sum(binomial(n-1,i)*a(n-i-1),i,1,n-1); /* Vladimir Kruchinin, Feb 22 2015 */

a(n) = if(n<2, n==0, subst( polinterpolate( Vec( serlaplace( exp( exp( x+O(x^n)/x )-1 ) ) ) ), x, n) )

from itertools import accumulate, islice
def A000296_gen():
    yield from (1,0)
    blist, a, b = (1,), 0, 1
    while True:
        blist = list(accumulate(blist, initial = (b:=blist[-1])))
        yield (a := b-a)
A000296_list = list(islice(A000296_gen(),20)) # Chai Wah Wu, Jun 22 2022

E.g.f.: exp(exp(x) - 1 - x).
B(n) = a(n) + a(n+1), where B = A000110 = Bell numbers [Becker].
Inverse binomial transform of Bell numbers (A000110).
a(n)= Sum_{k>=-1} (k^n/(k+1)!)/exp(1). - Vladeta Jovovic and Karol A. Penson, Feb 02 2003
a(n) = Sum_{k=0..n} ((-1)^(n-k))*binomial(n, k)*Bell(k) = (-1)^n + Bell(n) - A087650(n), with Bell(n) = A000110(n). - Wolfdieter Lang, Dec 01 2003
O.g.f.: A(x) = 1/(1-0*x-1*x^2/(1-1*x-2*x^2/(1-2*x-3*x^2/(1-... -(n-1)*x-n*x^2/(1- ...))))) (continued fraction). - Paul D. Hanna, Jan 17 2006
a(n) = Sum_{k = 0..n} {(-1)^(n-k) * Sum_{j = 0..k}((-1)^j * binomial(k,j) * (1-j)^n)/ k!} = sum over row n of A105794. - Tom Copeland, Jun 05 2006
a(n) = (-1)^n + Sum_{j=1..n} (-1)^(j-1)*B(n-j), where B(q) are the Bell numbers (A000110). - Emeric Deutsch, Oct 29 2006
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 >= 2, a(n) = (-1)^(n)charpoly(A,1). - Milan Janjic, Jul 08 2010
From Sergei N. Gladkovskii, Sep 20 2012, Oct 11 2012, Dec 19 2012, Jan 15 2013, May 13 2013, Jul 20 2013, Oct 19 2013, Jan 25 2014: (Start)
Continued fractions:
G.f.: (2/E(0) - 1)/x where E(k) = 1 + 1/(1 + 2*x/(1 - 2*(k+1)*x/E(k+1))).
G.f.: 1/U(0) where U(k) = 1 - x*k - x^2*(k+1)/U(k+1).
G.f.: G(0)/(1+2*x) where G(k) = 1 - 2*x*(k+1)/((2*k+1)*(2*x*k-x-1) - x*(2*k+1)*(2*k+3)*(2*x*k-x-1)/(x*(2*k+3) - 2*(k+1)*(2*x*k-1)/G(k+1))).
G.f.: (G(0) - 1)/(x-1) where G(k) = 1 - 1/(1+x-k*x)/(1-x/(x-1/G(k+1))).
G.f.: 1 + x^2/(1+x)/Q(0) where Q(k) = 1-x-x/(1-x*(2*k+1)/(1-x-x/(1-x*(2*k+2)/Q(k+1)))).
G.f.: 1/(x*Q(0)) where Q(k) = 1 + 1/(x + x^2/(1 - x - (k+1)/Q(k+1))).
G.f.: -(1+(2*x+1)/G(0))/x where G(k) = x*k - x - 1 - (k+1)*x^2/G(k+1).
G.f.: T(0) where T(k) = 1 - x^2*(k+1)/( x^2*(k+1) - (1-x*k)*(1-x-x*k)/T(k+1)).
G.f.: (1 + x * Sum_{k>=0} (x^k / Product_{p=0..k}(1 - p*x))) / (1 + x). (End)
a(n) = Sum_{i=1..n-1} binomial(n-1,i)*a(n-i-1), a(0)=1. - Vladimir Kruchinin, Feb 22 2015
G.f. A(x) satisfies: A(x) = (1/(1 + x)) * (1 + x * A(x/(1 - x)) / (1 - x)). - Ilya Gutkovskiy, May 21 2021
a(n) ~ exp(n/LambertW(n) - n - 1) * n^(n-1) / (sqrt(1 + LambertW(n)) * LambertW(n)^(n-1)). - Vaclav Kotesovec, Jun 28 2022

More terms, new description from Christian G. Bower, Nov 15 1999

a(23) corrected by Sean A. Irvine, Jun 22 2015

A036040 Irregular triangle of multinomial coefficients, read by rows (version 1).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 4, 3, 6, 1, 1, 5, 10, 10, 15, 10, 1, 1, 6, 15, 10, 15, 60, 15, 20, 45, 15, 1, 1, 7, 21, 35, 21, 105, 70, 105, 35, 210, 105, 35, 105, 21, 1, 1, 8, 28, 56, 35, 28, 168, 280, 210, 280, 56, 420, 280, 840, 105, 70, 560, 420, 56, 210, 28, 1, 1, 9, 36, 84, 126, 36, 252

Offset: 1

N. J. A. Sloane

This is different from A080575 and A178867.
T(n,m) = count of set partitions of n with block lengths given by the m-th partition of n.
From Tilman Neumann, Oct 05 2008: (Start)
These are also the coefficients occurring in complete Bell polynomials, Faa di Bruno's formula (in its simplest form) and computation of moments from cumulants.
Though the Bell polynomials seem quite unwieldy, they can be computed easily as the determinant of an n-dimensional square matrix. (See, e.g., Coffey (2006) and program below.)
The complete Bell polynomial of the first n primes gives A007446. (End)
From Tom Copeland, Apr 29 2011: (Start)
A relation between partition polynomials formed from these "refined" Stirling numbers of the second kind and umbral operator trees and Lagrange inversion is presented in the link "Lagrange a la Lah".
For simple diagrams of the relation between connected graphs, cumulants, and A036040, see the references on statistical physics below. In some sense, these graphs are duals of the umbral bouquets presented in "Lagrange a la Lah". (End)
These M3 (Abramowitz-Stegun) partition polynomials are the complete Bell polynomials (see a comment above) with recurrence (see the Wikipedia link) B_0 = 1, B_n = Sum_{k=0..n-1} binomial(n-1,k) * B_{n-1-k}*x[k+1], n >= 1. - Wolfdieter Lang, Aug 31 2016
With the indeterminates (x_1, x_2, x_3,...) = (t, -c_2*t, -c_3*t, ...) with c_n > 0, umbrally B(n,a.) =  B(n,t)|{t^n = a_n} = 0 and B(j,a.)B(k,a.) = B(j,t)B(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 A134685. - Tom Copeland, Feb 09 2018
For applications to functionals in quantum field theory, see Figueroa et al., Brouder, Kreimer and Yeats, and Balduf. In the last two papers, the Bell polynomials with the indeterminates (x_1, x_2, x_3,...) = (c_1, 2!c_2, 3!c_3, ...) are equivalent to the partition polynomials of A130561 in the indeterminates c_n. - Tom Copeland, Dec 17 2019
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 below. (End)
From Tom Copeland, Jun 12 2021: (Start)
These Bell polynomials and their relations to the Faa di Bruno Hopf bialgebra, correlation functions in quantum field theory, and the moment-cumulant duality are given on pp. 134 -144 of Zeidler.
An interpretation of the coefficients of the polynomials is given in expositions of the exponential formula, or principle, in Cameron et al., Duchamp, Duchamp et al., Labelle and Leroux, and Scott and Sokal along with some history. The simplest applications of this principle are given in A060540. (End)

			Triangle begins:
  1;
  1,  1;
  1,  3,  1;
  1,  4,  3,  6,  1;
  1,  5, 10, 10, 15, 10,  1;
  1,  6, 15, 10, 15, 60, 15, 20, 45, 15, 1;
  ...
The first partition of 3 (i.e., (3)) induces the set {{1, 2, 3}}, so T(3, 1) = 1; the second one (i.e., (2, 1)) the sets {{1, 2}, {3}}, {{1, 3}, {2}}, and {{2, 3}, {1}}, so T(3, 2) = 3; and the third one (i.e., (1, 1, 1)) the set {{1}, {2}, {3}}, so T(3, 1) = 1. - _Lorenzo Sauras Altuzarra_, Jun 20 2022

Abramowitz and Stegun, Handbook, p. 831, column labeled "M_3".
C. Itzykson and J. Drouffe, Statistical Field Theory Vol. 2, Cambridge Univ. Press, 1989, page 412.
S. Ma, Statistical Mechanics, World Scientific, 1985, page 205.
E. Zeidler, Quantum Field Theory II: Quantum Electrodynamics, Springer, 2009.

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.
P. Balduf, The propagator and diffeomorphisms of an interacting field theory, Master's thesis, submitted to the Institut für Physik, Mathematisch-Naturwissenschaftliche Fakultät, Humboldt-Universtität, Berlin, 2018.
F. Brglez, Of n-dimensional Dice, Combinatorial Optimization, and Reproducible Research: An Introduction, Elektrotehniski Vestnik, 78(4): 181-192, 2011.
P. Cameron, C. Krattenthaler, and T. Muller, Decomposable functors and the exponential principle, ll, arXiv:0911.3760 [math.CO], 2011.
Mark W. Coffey, A Set of Identities for a Class of Alternating Binomial Sums Arising in Computing Applications, arXiv:math-ph/0608049, 2006.
Tom Copeland, Lagrange a la Lah, Nov 11, 2011.
Tom Copeland, The creation / raising operators for Appell sequences, Nov 21, 2015.
Tom Copeland, Generators, Inversion, and Matrix, Binomial, and Integral Transforms, Dec 21, 2015.
Tom Copeland, Formal group laws and binomial Sheffer sequences, 2018.
G. Duchamp, Important formulas in combinatorics: The exponential formula, a Mathoverflow answer, 2015.
G. Duchamp, S. Goodenough, K. Penson, and L. Poinsot, Statistics on graphs, exponential formula, and combinatorial physics, arXiv:0910.0695 [cs.DM], 2010.
K. Ebrahimi-Fard and F. Patras, Cumulants, free cumulants and half-shuffles, arXiv:1409.5664v2 [math.CO], 2015, p. 16. [_Tom Copeland_, Feb 29 2016]
H. Figueroa and J. Gracia-Bondia, Combinatorial Hopf algebras in quantum field theory I, arXiv:0408145 [hep-th], 2005, (p. 41).
P. Guha, Riccati Chain, Higher Order Painlevé Type Equations and Stabilizer Set of Virasoro Orbit, 2006.
D. Kreimer and K. Yeats, Diffeomorphisms of quantum fields, arXiv:1610.01837 [math-ph], 2016. [_Tom Copeland_, Nov 23 2016]
G. Labelle and P. Leroux, An extension of the exponential formula in enumerative combinatorics , The Electron. J. Comb., Vol. 3, Issue 2, 1996.
Wolfdieter Lang, First 10 rows and polynomials
J. Novak and M. LaCroix, Three lectures on free probability, arXiv:1205.2097 [math.CO], 2012.
A. Scott and A. Sokal, Some variants of the exponential formula with applications to the multivariate Tutte polynomials (alias Potts model), arXiv:0803.1477 [math.CO], 2009.
J. Taylor, Formal group laws and hypergraph colorings, doctoral thesis, Univ. of Wash., 2016, p. 95. [_Tom Copeland_, Dec 20 2018]
Wikipedia, Bell polynomials

See A080575 for another version.
Row sums are the Bell numbers A000110.
Cf. A036036, A036037, A036038, A036039.
Cf. A000040, A007446, A178866 and A178867 (version 3).
Cf. A130561, A263634, A263916, A134685.
Cf. A127671.
Cf. A060540 for the coefficients of the compositions e^{ x^m/m! }.

with(combinat): nmax:=8: 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 sA036040(n,m):= n!/(mul((t!)^q(t)*q(t)!,t=1..n)); od: od: seq(seq(A036040(n, m), m=1..numbpart(n)), n=1..nmax); # Johannes W. Meijer, Jun 21 2010, Jul 12 2016

runs[li:{__Integer}] := ((Length/@ Split[ # ]))&[Sort@ li]; Table[temp=Map[Reverse, Sort@ (Sort/@ IntegerPartitions[w]), {1}]; Apply[Multinomial, temp, {1}]/Apply[Times, (runs/@ temp)!, {1}], {w, 6}]

completeBellMatrix := proc(x,n) // x - vector x[1]...x[m], m>=n
local i,j,M; begin
M := matrix(n,n): // zero-initialized
for i from 1 to n-1 do M[i,i+1] := -1: end_for:
for i from 1 to n do for j from 1 to i do
    M[i,j] := binomial(i-1,j-1)*x[i-j+1]: end_for: end_for:
return (M): end_proc:
completeBellPoly := proc(x, n) begin
return (linalg::det(completeBellMatrix (x,n))): end_proc:
for i from 1 to 10 do print(i, completeBellPoly(x,i)): end_for:
// Tilman Neumann, Oct 05 2008

A036040_poly(n,V=vector(n,i,eval(Str('x,i))))={matdet(matrix(n,n,i,j,if(j<=i,binomial(i-1,j-1)*V[n-i+j],-(j==i+1))))} \\ Row n of the sequence is made of the coefficients of the monomials ordered by increasing total order (sum of powers) and then lexicographically. - M. F. Hasler, Nov 16 2013, updated Jul 12 2014

from collections import Counter
def ASPartitions(n, k):
    Q = [p.to_list() for p in Partitions(n, length=k)]
    for q in Q: q.reverse()
    return sorted(Q)
def A036040_row(n):
    h = lambda p: product(map(factorial, Counter(p).values()))
    return [multinomial(p)//h(p) for k in (0..n) for p in ASPartitions(n, k)]
for n in (1..10): print(A036040_row(n))
# Peter Luschny, Dec 18 2016, corrected Apr 30 2022

E.g.f.: A(t) = exp(Sum_{k>=1} x[k]*(t^k)/k!).
T(n,m) is the coefficient of ((t^n)/n!)* x[1]^e(m,1)*x[2]^e(m,2)*...*x[n]^e(m,n) in A(t). Here the m-th partition of n, counted in Abramowitz-Stegun(A-St) order, is [1^e(m,1), 2^e(m,2), ..., n^e(m,n)] with e(m,j) >= 0 and if e(m, j)=0 then j^0 is not recorded.
a(n, m) = n!/Product_{j=1..n} j!^e(m,j)*e(m,j)!, with [1^e(m,1), 2^e(m,2), ..., n^e(m, n)] the m-th partition of n in the mentioned A-St order.
With the notation in the Lang reference, x(1) treated as a variable and D the derivative w.r.t. x(1), a raising operator for the polynomial S(n,x(1)) = P3_n(x[1], ..., x[n]) is R = Sum_{n>=0} x(n+1) D^n / n! ; i.e., R S(n, x(1)) = S(n+1, x(1)). The lowering operator is D; i.e., D S(n, x(1)) = n S(n-1, x(1)). The sequence of polynomials is an Appell sequence, so [S(.,x(1)) + y]^n = S(n, x(1) + y). For x(j) = (-1)^(j-1)* (j-1)! for j > 1, S(n, x(1)) = [x(1) - 1]^n + n [x(1) - 1]^(n-1). - Tom Copeland, Aug 01 2008
Raising and lowering operators are given for the partition polynomials formed from A036040 in the link in "Lagrange a la Lah Part I" on page 22. - Tom Copeland, Sep 18 2011
The n-th row is generated by the determinant of [Sum_{k=0..n-1} (x_(k+1)*(dP_n)^k/k!) - S_n], where dP_n is the n X n submatrix of A132440 and S_n is the n X n submatrix of A129185. The coefficients are flagged by the partitions of n represented by the monomials in the indeterminates x_k. Letting all x_n = t, generates the Bell / Touchard / exponential polynomials of A008277. - Tom Copeland, May 03 2014
The partition polynomials of A036039 are obtained by substituting (n-1)! x[n] for x[n] in the partition polynomials of this entry. - Tom Copeland, Nov 17 2015
-(n-1)! F(n, B(1, x[1]), B(2, x[1], x[2])/2!, ..., B(n, x[1], ..., x[n])/n!) = x[n] extracts the indeterminates of the complete Bell partition polynomials B(n, x[1], ..., x[n]) of this entry, where F(n, x[1], ..., x[n]) are the Faber polynomials of A263916. (Compare with A263634.) - Tom Copeland, Nov 29 2015; Sep 09 2016
T(n, m) = A127671(n, m)/A264753(n, m), n >= 1 and 1 <= m <= A000041(n). - Johannes W. Meijer, Jul 12 2016
From Tom Copeland, Sep 07 2016: (Start)
From the connections among the elementary Schur polynomials and the partition polynomials of A130561, A036039 and this array, the partition polynomials of this array satisfy (d/d(x_m)) P(n, x_1, ..., x_n) = binomial(n,m) * P(n-m, x_1, ..., x_(n-m)) with P(k, x_1, ..., x_n) = 0 for k < 0.
Just as in the discussion and example in A130561, the umbral compositional inverse sequence is given by the sequence P(n, x_1, -x_2, -x_3, ..., -x_n).
(End)
The partition polynomials with an index shift can be generated by (v(x) + d/dx)^n v(x). Cf. Guha, p. 12. - Tom Copeland, Jul 19 2018

More terms from David W. Wilson

Additional comments from Wouter Meeussen, Mar 23 2003

A080575 Triangle of multinomial coefficients, read by rows (version 2).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 4, 3, 6, 1, 1, 5, 10, 10, 15, 10, 1, 1, 6, 15, 15, 10, 60, 20, 15, 45, 15, 1, 1, 7, 21, 21, 35, 105, 35, 70, 105, 210, 35, 105, 105, 21, 1, 1, 8, 28, 28, 56, 168, 56, 35, 280, 210, 420, 70, 280, 280, 840, 560, 56, 105, 420, 210, 28, 1, 1, 9, 36, 36, 84, 252, 84, 126, 504, 378, 756, 126, 315, 1260, 1260, 1890, 1260, 126, 280, 2520, 840, 1260, 3780, 1260, 84, 945, 1260, 378, 36, 1, 1, 10, 45, 45, 120, 360, 120, 210, 840, 630, 1260, 210

Offset: 1

Wouter Meeussen, Mar 23 2003

This is different from A036040 and A178867.
T[n,m] = count of set partitions of n with block lengths given by the m-th partition of n in the canonical ordering.
From Tilman Neumann, Oct 05 2008: (Start)
These are also the coefficients occurring in complete Bell polynomials, Faa di Bruno's formula (in its simplest form) and computation of moments from cumulants.
Though the Bell polynomials seem quite unwieldy, they can be computed easily as the determinant of an n-dimensional square matrix. (see e.g. [Coffey] and program below)
The complete Bell polynomial of the first n primes gives A007446. (End)
The difference with A036040 and A178867 lies in the ordering of the monomials. This sequence uses lexicographic ordering, while in A036040 the total order (power) of the monomials prevails (Abramowitz-Stegun style): e.g., in row 6 we have ...+ 15*x[3]*x[5] + 15*x[3]*x[6]^2 + 10*x[4]^2 +...; in A036040 the coefficient of x[3]*x[6]^2 would come after that of x[4]^2 because the total order is higher, here it comes before in view of the lexicographic order. - M. F. Hasler, Jul 12 2015

			For n=4 the 5 integer partitions in canonical ordering with corresponding set partitions and counts are:
   [4]       -> #{1234} = 1
   [3,1]     -> #{123/4, 124/3, 134/2, 1/234} = 4
   [2,2]     -> #{12/34, 13/24, 14/23} = 3
   [2,1,1]   -> #{12/3/4, 13/2/4, 1/23/4, 14/2/3, 1/24/3, 1/2/34} = 6
   [1,1,1,1] -> #{1/2/3/4} = 1
Thus row 4 is [1, 4, 3, 6, 1].
Triangle begins:
1;
1, 1;
1, 3,  1;
1, 4,  3,  6,  1;
1, 5, 10, 10, 15,  10,  1;
1, 6, 15, 15, 10,  60, 20, 15,  45,  15,  1;
1, 7, 21, 21, 35, 105, 35, 70, 105, 210, 35, 105, 105, 21, 1;
...
Row 4 represents 1*k(4)+4*k(3)*k(1)+3*k(2)^2+6*k(2)*k(1)^2+1*k(1)^4 and T(4,4)=6 since there are six ways of partitioning four labeled items into one part with two items and two parts each with one item.

See A036040 for the column labeled "M_3" in Abramowitz and Stegun, Handbook, p. 831.

Alois P. Heinz, Rows n = 1..26, flattened
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Berg, Kimmo Complexity of solution structures in nonlinear pricing, Ann. Oper. Res. 206, 23-37 (2013).
R. J. Cano, Sequencer program.
Mark W. Coffey, A Set of Identities for a Class of Alternating Binomial Sums Arising in Computing Applications, arXiv:math-ph/0608049, 2006.
Wikipedia, Cumulant.
Wikipedia, Bell polynomials

See A036040 for another version. Cf. A036036-A036039.
Row sums are A000110.
Row lengths are A000041.
Cf. A007446. - Tilman Neumann, Oct 05 2008
Cf. A178866 and A178867 (version 3). - Johannes W. Meijer, Jun 21 2010
Maximum value in row n gives A102356(n).

runs[li:{__Integer}] := ((Length/@ Split[ # ]))&[Sort@ li]; Table[Apply[Multinomial, IntegerPartitions[w], {1}]/Apply[Times, (runs/@ IntegerPartitions[w])!, {1}], {w, 6}]
(* Second program: *)
completeBellMatrix[x_, n_] := Module[{M, i, j}, M[, ] = 0; For[i=1, i <= n-1 , i++, M[i, i+1] = -1]; For[i=1, i <= n , i++, For[j=1, j <= i, j++, M[i, j] = Binomial[i-1, j-1]*x[i-j+1]]]; Array[M, {n, n}]]; completeBellPoly[x_, n_] := Det[completeBellMatrix[x, n]]; row[n_] := List @@ completeBellPoly[x, n] /. x[] -> 1 // Reverse; Table[row[n], {n, 1, 10}] // Flatten (* _Jean-François Alcover, Aug 31 2016, after Tilman Neumann *)
B[0] = 1;
B[n_] := B[n] = Sum[Binomial[n-1, k] B[n-k-1] x[k+1], {k, 0, n-1}]//Expand;
row[n_] := Reverse[List @@ B[n] /. x[_] -> 1];
Table[row[n], {n, 1, 10}] // Flatten (* Jean-François Alcover, Aug 10 2018, after Wolfdieter Lang *)

completeBellMatrix := proc(x,n) // x - vector x[1]...x[m], m>=n
local i,j,M; begin M:=matrix(n,n): // zero-initialized
for i from 1 to n-1 do M[i,i+1]:=-1: end_for:
for i from 1 to n do for j from 1 to i do
    M[i,j] := binomial(i-1,j-1)*x[i-j+1]:
end_for: end_for:
return (M): end_proc:
completeBellPoly := proc(x, n) begin
return (linalg::det(completeBellMatrix(x,n))): end_proc:
for i from 1 to 10 do print(i,completeBellPoly(x,i)): end_for:
// Tilman Neumann, Oct 05 2008

PARI
```
\\ See links.
```

A080575_poly(n,V=vector(n,i,eval(Str('x,i))))={matdet(matrix(n,n,i,j,if(j<=i,binomial(i-1,j-1)*V[n-i+j],-(j==i+1))))}
A080575_row(n)={(f(s)=if(type(s)!="t_INT",concat(apply(f,select(t->t,Vec(s)))),s))(A080575_poly(n))} \\ M. F. Hasler, Jul 12 2015

A178867 Irregular triangle read by rows: multinomial coefficients, version 3; alternatively, row n gives coefficients of the n-th complete exponential Bell polynomial B_n(x_1, x_2, ...) with monomials sorted into some unknown order.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 6, 4, 3, 1, 1, 10, 10, 15, 5, 10, 1, 1, 15, 20, 45, 15, 60, 6, 15, 15, 10, 1, 1, 21, 35, 105, 35, 210, 21, 105, 105, 70, 7, 105, 35, 21, 1, 1, 28, 56, 210, 70, 560, 56, 420, 420, 280, 28, 840, 280, 168, 8, 280, 210, 105, 56, 35, 28, 1

Offset: 1

Johannes W. Meijer and Manuel Nepveu (Manuel.Nepveu(AT)tno.nl), Jun 21 2010, Jun 24 2010

"Exponential" here means in contrast to "ordinary", cf. A263633 (see Comtet). "Standard order" means as produced by Maple's "sort" command. - N. J. A. Sloane, Oct 28 2015
From Petros Hadjicostas, May 27 2020: (Start)
According to the Maple help files for the "sort" command, polynomials in multiple variables are "sorted in total degree with ties broken by lexicographic order (this is called graded lexicographic order)."
Thus, for example, x_1^2*x_3 = x_1*x_1*x_3 > x_1*x_2*x_2 = x_1*x_2^2, while x_1^2*x_4 = x_1*x_1*x_4 > x_1*x_2*x_3.
It appears that the authors' n-th row does give the coefficients of the n-th complete exponential Bell polynomial. Starting with row 6, however, it is unknown in what order the monomials of the n-th complete exponential Bell polynomial follow. It is definitely not the standard order nor any other known order. (End)
This version of the multinomial coefficients was discovered while calculating the probability that two 23 year old chessplayers would play each other on their birthday during a Dutch Chess Championship. This unique encounter took place on Apr 05 2008. Its probability is 1 in 50000 years, see the Meijer-Nepveu article.
The third version of the multinomial coefficients can be constructed with the basic multinomial coefficients A178866; see the formulas below. These multinomial coefficients appear in the columns of the multinomial coefficient matrix MCM[n,m] (n >= 1 and m >= 1).
We observe that the sum of the C(m,n) coefficients follow the A000296(n) sequence. The missing C(m, n=1) corresponds to A000296(1) = 0.
The number of multinomial coefficients in a triangle row leads to the partition numbers A000041. The row sums lead to the Bell numbers A000110.

			The first few complete exponential Bell polynomials are:
(1) x[1];
(2) x[1]^2 + x[2];
(3) x[1]^3 + 3*x[1]*x[2] + x[3];
(4) x[1]^4 + 6*x[1]^2*x[2] + 4*x[1]*x[3] + 3*x[2]^2 + x[4];
(5) x[1]^5 + 10*x[1]^3*x[2] + 10*x[1]^2*x[3] + 15*x[1]*x[2]^2 + 5*x[1]*x[4] + 10*x[2]*x[3] + x[5];
(6) x[1]^6 + 15*x[1]^4*x[2] + 20*x[1]^3*x[3] + 45*x[1]^2*x[2]^2 + 15*x[1]^2*x[4] + 60*x[1]*x[2]*x[3] + 6*x[1]*x[5] + 15*x[2]^3 + 15*x[2]*x[4] + 10*x[3]^2 + x[6].
(7) x[1]^7 + 21*x[1]^5*x[2] + 35*x[1]^4*x[3] + 105*x[1]^3*x[2]^2 + 35*x[1]^3*x[4] + 210*x[1]^2*x[2]*x[3] + 21*x[1]^2*x[5] + 105*x[1]*x[2]^3 + 105*x[1]*x[2]*x[4] + 70*x[1]*x[3]^2 + 7*x[1]*x[6] + 105*x[2]^2*x[3] + 35*x[3]*x[4] + 21*x[2]*x[5] + x[7].
...
The first few rows of the triangle are
  1;
  1,  1;
  1,  3,  1;
  1,  6,  4,   3,  1;
  1, 10, 10,  15,  5,  10,  1;
  1, 15, 20,  45, 15,  60,  6,  15,  15, 10, 1;
  1, 21, 35, 105, 35, 210, 21, 105, 105, 70, 7, 105, 35, 21, 1;
  ...

L. Comtet, Advanced Combinatorics, Reidel, 1974, pp. 134, 307-310.
J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, Chapter 2, Section 8 and table on page 49.

Kevin Brown, Generalized Birthday Problem (N Items in M Bins), 1994-2010.
Paul E. Gunnells, Generalized Catalan numbers from hypergraphs, arXiv:2102.05121 [math.CO], 2021. Mentions this sequence p. 27.
J. W. Meijer and M. Nepveu, Euler's ship on the Pentagonal Sea, Acta Nova, 4(1) (2008), 176-187.
Jin Wang, Nonlinear Inverse Relations for Bell Polynomials via the Lagrange Inversion Formula, J. Int. Seq., Vol. 22 (2019), Article 19.3.8.

Cf. A036040 (version 1 of multinomial coefficients), A080575 (version 2).

Cf. A000041, A000110, A000296, A178866, A263633.

with(combinat): nmax:=11; A178866(1):=1: T:=1: for n from 1 to nmax do y(n):=numbpart(n): P(n):=sort(partition(n)): k:=0: for r from 1 to y(n) do if P(n)[r,1]>1 then k:=k+1; B(k):=P(n)[r]: fi; od: A002865(n):=k; for k from 1 to A002865(n) do s:=0: j:=1: while sA002865(n))], `>`): for k from 1 to A002865(n) do M3[n,k]:=a[k] od: for k from 1 to A002865(n) do T:=T+1: A178866(T):= M3[n,k]: od: od:
mmax:=nmax: n:=1: for m from 1 to mmax do MCM[n,m]:= A178866(n) od: n:=2: r:=1: for i from 2 to nmax do p:=A002865(i): r:=r+1: while p>0 do for m from 1 to mmax do MCM[n,m]:=A178866(n)*binomial(m,r) od: p:=p-1: n:=n+1: od: od: T:=0: for m from 1 to mmax do for n from 1 to numbpart(m) do T:=T+1; A178867(T):= MCM[n,m]; od: od; seq(A178867(n), n=1..T);
# To produce the complete exponential Bell polynomials in standard order, from N. J. A. Sloane, Oct 28 2015
M:=12;
EE:=add(x[i]*t^i/i!,i=1..M);
t1:=exp(EE);
t2:=series(t1,t,M);
Q:=k->sort(expand(k!*coeff(t2,t,k)));
for k from 1 to 8 do lprint(k,Q(k)); od;
# To produce the coefficients of the complete exponential Bell polynomials in standard order:
triangle := proc(numrows) local E, s, Q;
E := add(x[i]*t^i/i!, i=1..numrows);
s := series(exp(E), t, numrows+1);
Q := k -> sort(expand(k!*coeff(s, t, k)));
seq(print(coeffs(Q(k))), k=1..numrows) end:
triangle(6); # Peter Luschny, May 27 2020

G.f.: Exp(Sum_{i >= 1} x_i*t^i/i!) = 1 + Sum_{n >= 1} B_n(x_1, x_2,...)*t^n/n!. [Comtet, p. 134, Eq. [3b]. - N. J. A. Sloane, Oct 28 2015]
For m >= 1, the formulas for the first few matrix columns are:
MCM[1,m] = A178866(1)*C(m,0) = 1*C(m,0);
MCM[2,m] = A178866(2)*C(m,2) = 1*C(m,2);
MCM[3,m] = A178866(3)*C(m,3) = 1*C(m,3);
MCM[4,m] = A178866(4)*C(m,4) = 3*C(m,4) and
MCM[5,m] = A178866(5)*C(m,4) = 1*C(m,4);
MCM[6,m] = A178866(6)*C(m,5) = 10*C(m,5) and
MCM[7,m] = A178866(7)*C(m,5) = 1*C(m,5);
MCM[8,m] = A178866(8)*C(m,6) = 15*C(m,6) and
MCM[9,m] = A178866(9)*C(m,6) = 15*C(m,6) and
MCM[10,m] = A178866(10)*C(m,6) = 10*C(m,6) and
MCM[11,m] = A178866(11)*C(m,6) = 1*C(m,6).

Alternative definition as coefficients of complete Bell polynomials added by N. J. A. Sloane, Oct 28 2015

Various sections and name edited by Petros Hadjicostas, May 28 2020

cp's OEIS Frontend

A000296 Set partitions without singletons: number of partitions of an n-set into blocks of size > 1. Also number of cyclically spaced (or feasible) partitions.

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A036040 Irregular triangle of multinomial coefficients, read by rows (version 1).

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A080575 Triangle of multinomial coefficients, read by rows (version 2).

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A178867 Irregular triangle read by rows: multinomial coefficients, version 3; alternatively, row n gives coefficients of the n-th complete exponential Bell polynomial B_n(x_1, x_2, ...) with monomials sorted into some unknown order.

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A211350 Refined triangle A124323: T(n,k) is the number of partitions of an n-set that are of type k (k-th integer partition, defined by A194602).

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A211360 Top elements of triangle A211350.

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