

A036039


Triangle of multinomial coefficients read by rows.


50



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
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OFFSET

1,4


COMMENTS

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. AdamsWatters, 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 integroderivatives, involving the digamma function and the Riemann zeta function values at positive integers, and to the characteristic polynomial for the adjacency matrix of complete ngraphs 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(n1, 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(1b.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


REFERENCES

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. 183185.


LINKS

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 MathÃ©matiques, Volume 130, Issue 3, AprilMay 2006, Pages 179222.
A. Bouali, Faber polynomials CayleyHamilton equation and Newton symmetric functions, Bulletin des Sciences MathÃ©matiques, Volume 130, Issue 1, JanFeb 2006, Pages 4970.
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]
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 [hepth], 2015.


FORMULA

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 / (1q^n)^2, the partition polynomials give an expansion of the MacMahon function M(q) = prod(n>=1, 1/(1q^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]/(n1)! for x[n] in the partition polynomials of this entry.
CIP_n(tF(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)


EXAMPLE

1
1 1
2 3 1
6 8 3 6 1
24 30 20 20 15 10 1 ...


MATHEMATICA

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 *)


CROSSREFS

Cf. A036036, A036037, A036038, A036040.
Cf. A102189 (rows reversed).
Cf. A, A115131, A263916, A094587.
Sequence in context: A076631 A035485 A074306 * A092271 A054115 A100822
Adjacent sequences: A036036 A036037 A036038 * A036040 A036041 A036042


KEYWORD

nonn,easy,nice,tabf


AUTHOR

N. J. A. Sloane


EXTENSIONS

More terms from David W. Wilson


STATUS

approved



