A102189 Array of multinomial coefficients (row reversed order of table A036039).

1, 1, 1, 1, 1, 3, 2, 1, 6, 3, 8, 6, 1, 10, 15, 20, 20, 30, 24, 1, 15, 45, 40, 15, 120, 90, 40, 90, 144, 120, 1, 21, 105, 70, 105, 420, 210, 210, 280, 630, 504, 420, 504, 840, 720, 1, 28, 210, 112, 420, 1120, 420, 105, 1680, 1120, 2520, 1344, 1120, 1260, 3360, 4032, 3360

Offset

0,6

Comments

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

The sequence of row lengths is [1,2,3,5,7,11,15,...] = A000041(n), n>=1 (partition numbers).

Row n of this array gives the coefficients of the cycle index polynomial n!*Z(S_n) for the symmetric group S_n. For instance, Z(S_4)= (x[1]^4 + 6*x[1]^2*x[2] + 3*x[2]^2 + 8*x[1]*x[3] + 6*x[4])/4!. The partitions of 4 appear here in the reversed Abramowitz-Stegun order.

See the W. Lang link "Solution of Newton's Identities" and the Note added Jun 06 2007 in the link "More rows and S_n cycle index polynomials" for the appearance of the signed array. - Wolfdieter Lang, Aug 01 2013

Multiplying the values of row n by the corresponding values in row n of A110141, one obtains n!. - Jaimal Ichharam, Aug 06 2015

Links

Andrew Howroyd, Table of n, a(n) for n = 0..2713 (rows 0..20)

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, pp. 831-2.

Wolfdieter Lang, More rows and S_n cycle index polynomials.

Wolfdieter Lang, Solution of Newton's Identities.

Andrei Vieru, Analytic renormalization of multiple zeta functions. Geometry and combinatorics of generalized Euler reflection formula for MZV, arXiv preprint arXiv:1601.04703 [math.NT], 2016.

Example

Triangle begins:
  [1];
  [1];
  [1,1];
  [1,3,2];
  [1,6,3,8,6];
  [1,10,15,20,20,30,24];
  ...

Mathematica

aspartitions[n_] := Reverse /@ Sort[Sort /@ IntegerPartitions[n]]; ascycleclasses[n_Integer] := n!/(Times @@ #)& /@ ((#! Range[n]^#)& /@ Function[par, Count[par, #]& /@ Range[n]] /@ aspartitions[n]); row[n_] := ascycleclasses[n] // Reverse; Table[row[n], {n, 1, 8}] // Flatten (* Jean-François Alcover, Feb 04 2014, after A036039 and Wouter Meeussen *)

Prog

(PARI)
C(sig)={my(S=Set(sig)); vecsum(sig)!/(vecprod(sig)*prod(k=1, #S, (#select(t->t==S[k], sig))!))}
Row(n)={apply(C, [Vecrev(p) | p<-Vecrev(partitions(n))])}
{ for(n=0, 7, print(Row(n))) } \\ Andrew Howroyd, Oct 06 2025

Crossrefs

Cf. A000041, A036039, A110141.

Sequence in context: A134199 A323417 A379487 * A031252 A208152 A194761

Adjacent sequences: A102186 A102187 A102188 * A102190 A102191 A102192

Keyword

nonn,easy,tabf

Author

Wolfdieter Lang, Feb 15 2005

Extensions

a(0)=1 prepended by Andrew Howroyd, Oct 06 2025

Status

approved