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

%I A102189 #38 Apr 01 2025 03:38:30
%S A102189 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,
%T A102189 144,120,1,21,105,70,105,420,210,210,280,630,504,420,504,840,720,1,28,
%U A102189 210,112,420,1120,420,105,1680,1120,2520,1344,1120,1260,3360,4032,3360
%N A102189 Array of multinomial numbers (row reversed order of table A036039).
%C A102189 See Abramowitz and Stegun, Handbook, p. 831, column labeled "M_2", read backwards.
%C A102189 The sequence of row lengths is [1,2,3,5,7,11,15,...] = A000041(n), n>=1 (partition numbers).
%C A102189 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.
%C A102189 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
%C A102189 Multiplying the values of row n by the corresponding values in row n of A110141, one obtains n!. - _Jaimal Ichharam_, Aug 06 2015
%H A102189 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards Applied Math. Series 55, Tenth Printing, 1972, pp. 831-2.
%H A102189 Wolfdieter Lang, <a href="/A102189/a102189.pdf">More rows and S_n cycle index polynomials.</a>
%H A102189 Wolfdieter Lang, <a href="/A102189/a102189_1.pdf">Solution of Newton's Identities.</a>
%H A102189 Andrei Vieru, <a href="https://arxiv.org/abs/1601.04703">Analytic renormalization of multiple zeta functions. Geometry and combinatorics of generalized Euler reflection formula for MZV</a>, arXiv preprint arXiv:1601.04703 [math.NT], 2016.
%e A102189 Triangle begins:
%e A102189   [1];
%e A102189   [1,1];
%e A102189   [1,3,2];
%e A102189   [1,6,3,8,6];
%e A102189   [1,10,15,20,20,30,24];
%e A102189   ...
%t A102189 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_ *)
%Y A102189 Cf. A000041, A036039, A110141.
%K A102189 nonn,easy,tabf
%O A102189 1,5
%A A102189 _Wolfdieter Lang_, Feb 15 2005