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A036039 Triangle of multinomial coefficients read by rows. 61
1, 1, 1, 2, 3, 1, 6, 8, 3, 6, 1, 24, 30, 20, 20, 15, 10, 1, 120, 144, 90, 40, 90, 120, 15, 40, 45, 15, 1, 720, 840, 504, 420, 504, 630, 280, 210, 210, 420, 105, 70, 105, 21, 1, 5040, 5760, 3360, 2688, 1260, 3360, 4032, 3360, 1260, 1120, 1344, 2520, 1120, 1680, 105, 420 (list; graph; refs; listen; history; text; internal format)



The sequence of row lengths is A000041(n), n>=1, (partition numbers).

Number of permutations whose cycle structure is the given partition. Row sums are factorials (A000142). - Franklin T. Adams-Watters, Jan 12 2006

A relation between partition polynomials formed from these "refined" Stirling numbers of the first kind and umbral operator trees and Lagrange inversion is presented in the link "Lagrange a la Lah".

These cycle index polynomials for the symmetric group S_n are also related to a raising operator / infinitesimal generator for fractional integro-derivatives, involving the digamma function and the Riemann zeta function values at positive integers, and to the characteristic polynomial for the adjacency matrix of complete n-graphs A055137 (cf. MathOverflow link). - Tom Copeland, Nov 03 2012

In the Lang link, replace all x(n) by t to obtain A132393. Furthermore replace x(1) by t and all other x(n) by 1 to obtain A008290. See A274760. - Tom Copeland, Nov 06 2012, Oct 29 2015 - corrected by Johannes W. Meijer, Jul 28 2016

The umbral compositional inverses of these polynomials are formed by negating the indeterminates x(n) for n>1, i.e., P(n,P(.,x(1),-x(2),-x(3),..),x(2),x(3),..) = x(1)^n (cf. A130561 for an example of umbral compositional inversion). The polynomials are an Appell sequence in x(1), i.e., dP(n,x(1))/dx(1) = n P(n-1, x(1)) and (P(.,x)+y)^n=P(n,x+y) umbrally, with P(0,x(1))=1. - Tom Copeland, Nov 14 2014

Regarded as the coefficients of the partition polynomials listed by Lang, a signed version of these polynomials IF(n,b1,b2,...,bn) (n! times polynomial on page 184 of Airault and Bouali) provides an inversion of the Faber polynomials F(n,b1,b2,...,bn) (page 52 of Bouali, A263916, and A115131). For example,  F(3, IF(1,b1), IF(2,b1,b2)/2!, IF(3,b1,b2,b3)/3!) = b3 and IF(3, F(1,b1), F(2,b1,b2), F(3,b1,b2,b3))/3! = b3 with F(1,b1) = -b1. (Compare with A263634.) - Tom Copeland, Oct 28 2015; Sep 09 2016)

The e.g.f. for the row partition polynomials is Sum_{n>=0} P_n(b_1,..,b_n) x^n/n! = exp[Sum_{n>=1} b_n x^n/n], or, exp[P.(b_1,..,b_n)x] = exp[-<ln(1-b.x)>], expressed umbrally with <"power series"> denoting umbral evaluation (b.)^n = b_n within the power series. This e.g.f. is central to the paper by Maxim and Schuermannn on characteristic classes (cf. Friedrich and McKay also). - Tom Copeland, Nov 11 2015

The elementary Schur polynomials are given by S(n,x(1),x(2),..,x(n)) = P(n,x(1), 2*x(2),..,n*x(n)) / n!. See p. 12 of Carrell. - Tom Copeland, Feb 06 2016


Abramowitz and Stegun, Handbook, p. 831, column labeled "M_2".

Mohammad K. Azarian, Problem 1218, Pi Mu Epsilon Journal, Vol. 13, No. 2, Spring 2010, p. 116.  Solution published in Vol. 13, No. 3, Fall 2010, pp. 183-185.


Table of n, a(n) for n=1..60.

Milton Abramowitz and Irene A. Stegun, editors, Multinomials: M_1, M_2 and M_3, Handbook of Mathematical Functions, December 1972, pp. 831-2.

H. Airault and A. Bouali, Differential calculus on the Faber polynomials, Bulletin des Sciences Mathématiques, Volume 130, Issue 3, April-May 2006, Pages 179-222.

A. Bouali, Faber polynomials Cayley-Hamilton equation and Newton symmetric functions, Bulletin des Sciences Mathématiques, Volume 130, Issue 1, Jan-Feb 2006, Pages 49-70.

S. Carrell, Combinatorics of the KP Hierarchy, Thesis, University of Waterloo, Ontario, Canada, 2009

T. Copeland, Lagrange a la Lah, Riemann zeta function at positive integers and an Appell sequence of polynomials, The creation / raising operators for Appell sequences, Generators, Inversion, and Matrix, Binomial, and Integral Transforms, Addendum to Elliptic Lie Triad

Mark Dominus Cycle classes of permutations [From Wouter Meeussen, Jun 26 2009]

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

R. Friedrich and J. McKay, Formal groups, Witt vectors and free probablility, arXiv:1204.6522 [math.OA], 2012.

Wolfdieter Lang, First ten rows and polynomials.

L. Maxim and J. Schuermann, Equivariant characteristic classes of external and symmetric products of varieties, arXiv:1508.04356 [math.AG], 2015.

R. Szabo, N=2 guage theories, instanton moduli spaces and geometric representation theory, arXiv:1507.00685 [hep-th], 2015.


Raising and lowering operators are given for the partition polynomials formed from this sequence in the link in "Lagrange a la Lah Part I" on pg. 23. - Tom Copeland, Sep 18 2011

From Szabo p. 34, with b_n = q^n / (1-q^n)^2, the partition polynomials give an expansion of the MacMahon function M(q) = Product_{n>=1} 1/(1-q^n)^n = Sum_{n>=0} PL(n) q^n, the generating function for PL(n) = n! P_n(b_1,..,b_n), the number of plane partitions with sum n. - Tom Copeland, Nov 11 2015

From Tom Copeland, Nov 18 2015: (Start)

The partition polynomials of A036040 are obtained by substituting x[n]/(n-1)! for x[n] in the partition polynomials of this entry.

CIP_n(t-F(1,b1),-F(2,b1,b2),..,-F(n,b1,..,bn)) = P_n(b1,..,bn;t), where CIP_n are the partition polynomials of this entry; F(n,..), those of A263916; and P_n, those defined in my formula in A094587, e.g., P_2(b1,b2;t) = 2 b2 + 2 b1 t + t^2.

CIP_n(-F(1,b1),-F(2,b1,b2),..,-F(n,b1,..,bn)) = n! bn. (End)

From the relation to the elementary Schur polynomials given in A130561 and above, the partition polynomials of this array satisfy (d/d(x_m)) P(n,x_1,..,x_n) = (1/m) * (n!/(n-m)!) * P(n-m,x_1,..,x_(n-m)) with P(k,..) = 0 for k<0. - Tom Copeland, Sep 07 2016



   1  1

   2  3  1

   6  8  3  6  1

  24 30 20 20 15 10 1 ...


nmax:=7: with(combinat): 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: A036039(n, m) := n!/ (mul((t)^q(t)*q(t)!, t=1..n)); od: od: seq(seq(A036039(n, m), m=1..numbpart(n)), n=1..nmax); # Johannes W. Meijer, Jul 14 2016


aspartitions[n_]:=Reverse/@Sort[Sort/@IntegerPartitions[n]]; (* Abramowitz & Stegun ordering *);

ascycleclasses[n_Integer]:=n!/(Times@@ #)&/@((#!

Range[n]^#)&/@Function[par, Count[par, # ]&/@Range[n]]/@aspartitions[n])

(* The function "ascycleclasses" is then identical with A&S multinomial M2. *)

(* Wouter Meeussen, Jun 26 2009, Jun 27 2009 *)



def A036039_row(n):

    fn, C = factorial(n), []

    for k in (0..n):

        for p in Partitions(n, length=k):

            fp = 1; pf = 1

            for a, c in p.to_exp_dict().iteritems():

                fp *= factorial(c)

                pf *= factorial(a)**c

            co = fn//(fp*pf)

            C.append(co*prod([factorial(i-1) for i in p]))

    return C

[A036039_row(n) for n in (1..7)] # Peter Luschny, Feb 05 2016


Cf. A036036, A036037, A036038, A036040.

Cf. A102189 (rows reversed).

Cf. A115131, A263916, A094587, A130561, A263634.

Sequence in context: A076631 A035485 A074306 * A092271 A054115 A100822

Adjacent sequences:  A036036 A036037 A036038 * A036040 A036041 A036042




N. J. A. Sloane


More terms from David W. Wilson



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Last modified October 25 02:02 EDT 2016. Contains 277127 sequences.