A264753 -id:A264753 - cp's OEIS frontend

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

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

A127671 Cumulant expansion numbers: Coefficients in expansion of log(1 + Sum_{k>=1} x[k]*(t^k)/k!).

Original entry on oeis.org

1, 1, -1, 1, -3, 2, 1, -4, -3, 12, -6, 1, -5, -10, 20, 30, -60, 24, 1, -6, -15, -10, 30, 120, 30, -120, -270, 360, -120, 1, -7, -21, -35, 42, 210, 140, 210, -210, -1260, -630, 840, 2520, -2520, 720, 1, -8, -28, -56, -35, 56, 336, 560, 420, 560, -336, -2520, -1680, -5040, -630, 1680, 13440, 10080, -6720

Offset: 1

Wolfdieter Lang, Jan 23 2007

Connected objects from general (disconnected) objects.
The row lengths of this array is p(n):=A000041(n) (partition numbers).
In row n the partitions of n are taken in the Abramowitz-Stegun order.
One could call the unsigned numbers |a(n,k)| M_5 (similar to M_0, M_1, M_2, M_3 and M_4 given in A111786, A036038, A036039, A036040 and A117506, resp.).
The inverse relation (disconnected from connected objects) is found in A036040.
(d/da(1))p_n[a(1),a(2),...,a(n)] = n b_(n-1)[a(1),a(2),...,a(n-1)], where p_n are the partition polynomials of the cumulant generator A127671 and b_n are the partition polynomials for A133314. - Tom Copeland, Oct 13 2012
See notes on relation to Appell sequences in a differently ordered version of this array A263634. - Tom Copeland, Sep 13 2016
Given a binomial Sheffer polynomial sequence defined by the e.g.f. exp[t * f(x)] = Sum_{n >= 0} p_n(t) * x^n/n!, the cumulants formed from these polynomials are the Taylor series coefficients of f(x) multipied by t. An example is the sequence of the Stirling polynomials of the first kind A008275 with f(x) = log(1+x), so the n-th cumulant is (-1)^(n-1) * (n-1)! * t. - Tom Copeland, Jul 25 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 in A036040. (End)
Ignoring signs, these polynomials appear in Schröder in the set of equations (II) on p. 343 and in Stewart's translation on p. 31. - Tom Copeland, Aug 25 2021

			Row n=3: [1,-3,2] stands for the polynomial 1*x[3] - 3*x[1]*x[2] + 2*x[1]^3 (the Abramowitz-Stegun order of the p(3)=3 partitions of n=3 is [3],[1,2],[1^3]).

C. Itzykson and J.-M. Drouffe, Statistical field theory, vol. 2, p. 413, eq.(13), Cambridge University Press, (1989).

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, pp. 831-2.
Tom Copeland, The creation / raising operators for Appell sequences.
Wolfdieter Lang, First 10 rows of cumulant numbers and polynomials
E. Schröder, Ueber unendlich viele Algorithmen zur Auflösung der Gleichungen, Mathematische Annalen vol. 2, 317-365, 1870.
G. Stewart, On infinitely many algorithms for solving equations, 1993, (translation into English of Schröder's paper above).

Cf. A133314, A263916, A263634.
Cf. A008275.
Cf. A036039, A036040, A130561.

E.g.f. for multivariate row polynomials A(t) := log(1 + Sum_{k>=1} x[k]*(t^k)/k!).
Row n polynomial p_n(x[1],...,x[n]) = [(t^n)/n!]A(t).
a(n,m) = A264753(n, m) * M_3(n,m) with M_3(n,m) = A036040(n,m) (Abramowitz-Stegun M_3 numbers). - corrected by Johannes W. Meijer, Jul 12 2016
p_n(x[1],...,x[n]) = -(n-1)!*F(n,x[1],x[2]/2!,..,x[n]/n!) in terms of the Faber polynomials F(n,b1,..,bn) of A263916. - Tom Copeland, Nov 17 2015
With D = d/dz and M(0) = 1, the differential operator R = z + d(log(M(D))/dD = z + d(log(1 + x[1] D + x[2] D^2/2! + ...))/dD = z + p.*exp(p.D) = z + Sum_{n>=0} p_(n+1)(x[1],..,x[n]) D^n/n! is the raising operator for the Appell sequence A_n(z) = (z + x[.])^n = Sum_{k=0..n} binomial(n,k) x[n-k] z^k with the e.g.f. M(t) e^(zt), i.e., R A_n(z) = A_(n+1)(z) and dA_n(z)/dz = n A_(n-1)(z). The operator Q = z - p.*exp(p.D) generates the Appell sequence with e.g.f. e^(zt) / M(t). - Tom Copeland, Nov 19 2015

cp's OEIS Frontend

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

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