cp's OEIS Frontend

a(n) depends only on the prime signature of n (cf. A025487). So a(24) = a(375) since 24 = 2^3 * 3 and 375 = 3 * 5^3 both have prime signature (3,1).

Multinomial coefficients in prime factorization order. - Max Alekseyev, Nov 07 2006

The Dirichlet inverse is given by A080339, negating all but the A080339(1) element in A080339. - R. J. Mathar, Jul 15 2010

Number of (distinct) permutations of the multiset of prime factors. - Joerg Arndt, Feb 17 2015

Number of not divisible chains in the divisor lattice of n. - Peter Luschny, Jun 15 2013

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)

This array gives in row n>=1 the multinomial numbers (call them M_0 numbers) m!/product((a_j)!,j=1..n) with the exponents of the partitions of n with number of parts m:=sum(a_j,j=1..n), given in the Abramowitz-Stegun order. See p. 831 of the given reference. See also the arrays for the M_1, M_2 and M_3 multinomial numbers A036038, A036039 and A036040 (or A080575).

For a signed version see A111786.

These M_0 multinomial numbers give the number of compositions of n >= 1 with parts corresponding to the partitions of n (in A-St order). See an n = 5 example below. The triangle with the summed entries of like number of parts m is A007318(n-1, m-1) (Pascal). - Wolfdieter Lang, Jan 29 2021

The upper left corner of the array is T(0,1).

Without loss of generality, the permutation can be taken to be (1 2 ... k) (k+1 k+2 ... 2k) ... ((n-1)k+1 (n-1)k+2 ... nk).

Note that it is the partition that is invariant, not the individual parts. Thus for n=k=2 with permutation (1 2)(3 4), the partition 1,3|2,4 is counted; it maps to 2,4|1,3, which is the same partition.

From Tilman Neumann, Oct 05 2008: (Start)

a(n) is also given by

- substituting the primes (A000040) into (the simplest) Faa di Bruno's formula, or

- the complete Bell polynomial of the first n prime arguments, or

- computing n-th moments from the first n primes as cumulants

The examples show that the coefficients of the prime power products are just A036040/A080575 (these are just rearrangements of the same coefficients). Moreover, the prime products of the additional terms span the whole space of natural numbers, thus what we see here is a reordering of the natural numbers! (End)

"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.

A set partition of m-shape is a partition of a set with cardinality m*n for some n >= 0 such that the sizes of the blocks are m times the parts of the integer partitions of n.

If m = 0, all possible sizes are zero. Thus the number of set partitions of 0-shape is the number of integer partitions of n (partition numbers A000041).

If m = 1, the set is {1, 2, ..., n} and the set of all possible sizes are the integer partitions of n. Thus the number of set partitions of 1-shape is the number of set partitions (Bell numbers A000110).

If m = 2, the set is {1, 2, ..., 2n} and the number of set partitions of 2-shape is the number of set partitions into even blocks A005046.

From Petros Hadjicostas, Aug 06 2019: (Start)

Irwin (1916) proved the following combinatorial result: Assume r_1, r_2, ..., r_n are positive integers and we have r_1*r_2*...*r_n objects. We divide them into r_1 classes of r_2*r_3*...*r_n objects each, then each class into r_2 subclasses of r_3*...*r_n objects each, and so on. We call each such classification, without reference to order, a "classification" par excellence. He proved that the total number of classifications is (r_1*r_2*...*r_n)!/( r1! * (r_2!)^(r_1) * (r_3!)^(r_1*r_2) * ... (r_n!)^(r_1*r_2*...*r_{n-1}) ).

Apparently, this problem appeared in Carmichael's "Theory of Numbers".

This result can definitely be used to prove some special cases of my conjecture below. (End)

A generalization of the triangle of Stirling numbers of the second kind, these are the coefficients appearing in the expansion of (x_1 + x_2 + x_3 + ...)^n in terms of augmented monomial symmetric functions. They also appear in Faa di Bruno's formula.

a(n) is the maximum value in row n of A080575.

All multinomial coefficients (MC's) are equal, but some are more equal than others. These are the basic multinomial coefficients (BMC's). The ordinary multinomial coefficients can be generated with the basic multinomial coefficients; see A178867.

A number n can be partitioned in A000041(n) different ways. The seven partitions of n=5 are e.g. [5] = [1+4] = [2+3] = [1+1+3] = [1+2+2] = [1+1+1+2] = [1+1+1+1+1]. We observe that the k-th partition of n will consist of a certain number of 1s (i.e., "singles"), a certain number of 2s (i.e., "pairs"), a certain number of 3s (i.e., "triples"), a certain number of 4s (i.e., "4-tuples") and so on. We denote with qt the number of t-tuples in the k-th partition of n. We observe that for the third partition of n=5 there is one pair (q2=1) and one triple (q3=1).

The multinomial coefficients are defined by M3[n,k] = n!/product((t!)^qt*(qt)!, t = 1..n), see Abramowitz and Stegun. For the third partition M3[5,3] = 10, so there are 10 different ways of distributing one pair (B1, B1) and one triple (B2, B2, B2) over five positions.

We define the BMC's as the multinomial coefficients M3[n,k] for which there are no singles (q1=0) in the k-th partition of n for n>=2. Furthermore we define a(1) = 1.

The number of a(n) terms in a triangle row lead to sequence A002865(n) (n>=2). The row sums lead to sequence A000296(n) (n>=2).