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A036040 Irregular triangle of multinomial coefficients, read by rows (version 1). 94
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 (list; graph; refs; listen; history; text; internal format)



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] and program below.)

The complete Bell polynomial of the first n primes gives A007446.


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". - Tom Copeland, Apr 29 2011

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


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.


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.

F. Brglez, Of n-dimensional Dice, Combinatorial Optimization, and Reproducible Research: An Introduction, Elektrotehniski Vestnik, 78(4): 181-192, 2011.

Mark W. Coffey, A Set of Identities for a Class of Alternating Binomial Sums Arising in Computing Applications, arXiv:math-ph/0608049, 2006.

T. Copeland, Lagrange a la Lah, Nov 11, 2011.

T. Copeland, The creation / raising operators for Appell sequences, Nov 21, 2015.

T. Copeland, Generators, Inversion, and Matrix, Binomial, and Integral Transforms, Dec 21, 2015.

G. Duchamp, Important formulas in combinatorics: The exponential formula, a Mathoverflow answer, 2015.

K. Ebrahimi-Fard and F. Patras, Cumulants, free cumulants and half-shuffles, arXiv:1409.5664v2 [math.CO], 2015, p. 16.

D. Kreimer and K. Yeats, Diffeomorphisms of quantum fields, arXiv:1610.01837 [math-ph], 2016.

Wolfdieter Lang, First 10 rows and polynomials

Wikipedia, Bell polynomials


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!^e(m,j))*e(m,j)!,j=1..n ), 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=0,1,...) 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,..) = 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).




1,  1;

1,  3,  1;

1,  4,  3,  6,  1;

1,  5, 10, 10, 15, 10,  1;


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 s<n do j:=j+1: s:=s+B(m)[j]: x(j):=B(m)[j]: end do; jmax:=j; for r from 1 to n do q(r):=0 od: for r from 1 to n do for j from 1 to jmax do if x(j)=r then q(r):=q(r)+1 fi: od: od: A036040(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

(PARI) 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


def A036040_row(n):

    fn, C = factorial(n), []

    # Function PartAS(n) defined in A036039.

    for q in PartAS(n):


        p = Partition(q)

        factorial_product = 1

        power_factorial_product = 1

        for part, count in p.to_exp_dict().iteritems():

            factorial_product *= factorial(count)

            power_factorial_product *= factorial(part)**count


    return C

for n in (1..10): print A036040_row(n) # Peter Luschny, Dec 18 2016


See A080575 for another version. Cf. A036036-A036039.

Row sums are the Bell numbers A000110.

Cf. A000040, A007446.

Cf. A178866 and A178867 (version 3).

Cf. A130561, A263634, A263916.

Sequence in context: A126015 A247169 A144336 * A080575 A205117 A077228

Adjacent sequences:  A036037 A036038 A036039 * A036041 A036042 A036043




N. J. A. Sloane


More terms from David W. Wilson

Additional comments from Wouter Meeussen, Mar 23 2003



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Last modified August 17 13:03 EDT 2017. Contains 290635 sequences.