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A036039 Triangle of multinomial coefficients read by rows. 46
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 A008290. - Tom Copeland, Nov 06 2012 (Corrected Oct 29 2015)

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. 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. - Tom Copeland, Oct 28 2015

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


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.

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].

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.

Tom Copeland, Lagrange a la Lah

Tom Copeland, Riemann zeta function at positive integers and an Appell sequence of polynomials

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

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) = prod(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






24,30,20,20,15,10,1; ...


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 *)


Cf. A036036, A036037, A036038, A036040.

Cf. A102189 (rows reversed).

Cf. A, A115131, A263916.

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 November 28 08:05 EST 2015. Contains 264557 sequences.