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A036040 Triangle of multinomial coefficients, read by rows (version 1). 80
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.

Contribution 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]


Abramowitz and Stegun, Handbook, p. 831, column labeled "M_3".

F. Brglez, Of n-dimensional Dice, Combinatorial Optimization, and Reproducible Research: An Introduction, ELEKTROTEHNISKI VESTNIK 78(4): 181-192, 2011; http://ev.fe.uni-lj.si/4-2011/Brglez.pdf

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.


T. D. Noe, Rows n=1..25 of triangle, 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].

Mark W. Coffey, A Set of Identities for a Class of Alternating Binomial Sums Arising in Computing Applications

T. Copeland, Lagrange a la Lah

Wolfdieter Lang, First 10 rows and polynomials

Wikipedia, Bell polynomials


E.g.f. A(t)= exp(sum(x[k]*(t^k)/k!,k=1..infinity)).

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








with(combinat): nmax:=8: T:=0: for n from 1 to nmax do y(n):=numbpart(n): P(n):=sort(partition(n)): for r from 1 to y(n) do B(r):=P(n)[r] od: for k from 1 to y(n) do s:=0: j:=0: while s<n do j:=j+1: s:=s+B(k)[j]: x(j):=B(k)[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: M3[n, k]:= n!/(mul((t!)^q(t)*q(t)!, t=1..n)); od: for k from 1 to y(n) do T:= T+1: A036040(T):= M3[n, k]: od: od: seq(A036040(n), n=1..T); [Johannes W. Meijer, Jun 21 2010]


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}]


Contribution from Tilman Neumann, Oct 05 2008: (Start)


completeBellMatrix := proc(x, n)

// x - vector x[1]...x[m], m>=n

local i, j, M;


M:=matrix(n, n): // zero-initialized

for i from 1 to n-1 do

M[i, i+1]:=-1:


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]:



return (M):


completeBellPoly := proc(x, n)


return (linalg::det(completeBellMatrix(x, n))):


for i from 1 to 10 do print(i, completeBellPoly(x, i)): end_for:


(PARI) A036040_poly(n)=matdet(matrix(n, n, i, j, if(j<=i, binomial(i-1, j-1)*eval(Str("X", i-j+1)), -(j==i+1)))) \\ - M. F. Hasler, Nov 16 2013


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

Row sums are the Bell numbers A000110.

Cf. A000040, A007446.

Cf. A178866 and A178867 (version 3).

Sequence in context: A049999 A126015 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 April 18 05:34 EDT 2014. Contains 240688 sequences.