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A036039 Triangle of multinomial coefficients read by rows. 45
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)
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. 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 x(1) by x and all other x(n) by 1 to obtain

A008290. - Tom Copeland, Nov 06 2012

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. 183-185.

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].

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]

Wolfdieter Lang, First ten rows and polynomials.

FORMULA

Raising and lowering operators are given for the partition polynomials formed from A036039 in the link in "Lagrange a la Lah Part I" on pg. 23. - Tom Copeland, Sep 18 2011

EXAMPLE

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

MATHEMATICA

Mathematica from Wouter Meeussen, Jun 26 2009, Jun 27 2009 (Start)

aspartitions[n_]:=Reverse/@Sort[Sort/@Partitions[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. (End)

CROSSREFS

Cf. A036036-A036040.

Cf. A102189 (rows reversed).

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

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Last modified April 23 11:09 EDT 2014. Contains 240919 sequences.