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A036040 Triangle of multinomial coefficients, read by rows (version 1). 84
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)
OFFSET

1,5

COMMENTS

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.

(End)

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

REFERENCES

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.

LINKS

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].

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

T. Copeland, Lagrange a la Lah

Wolfdieter Lang, First 10 rows and polynomials

Wikipedia, Bell polynomials

FORMULA

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

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 sub-matrix of A132440 and S_n is the n X n sub-matrix 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

EXAMPLE

1;

1,1;

1,3,1;

1,4,3,6,1;

1,5,10,10,15,10,1;

MAPLE

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

MATHEMATICA

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}]

PROG

From Tilman Neumann, Oct 05 2008: (Start)

(MuPAD)

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:

(End)

(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

CROSSREFS

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: A126015 A247169 A144336 * A080575 A205117 A077228

Adjacent sequences:  A036037 A036038 A036039 * A036041 A036042 A036043

KEYWORD

nonn,easy,nice,tabf,look

AUTHOR

N. J. A. Sloane

EXTENSIONS

More terms from David W. Wilson

Additional comments from Wouter Meeussen, Mar 23 2003

STATUS

approved

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Last modified December 22 07:03 EST 2014. Contains 252328 sequences.