cp's OEIS Frontend

A324254 Signed version of the partition array A036039 (signed M_2 multinomial numbers).

%I A324254 #16 Dec 15 2024 15:36:32
%S A324254 1,-1,1,2,-3,1,-6,8,3,-6,1,24,-30,-20,20,15,-10,1,-120,144,90,40,-90,
%T A324254 -120,-15,40,45,-15,1,720,-840,-504,-420,504,630,280,210,-210,-420,
%U A324254 -105,70,105,-21,1,-5040,5760,3360,2688,1260,-3360,-4032,-3360,-1260,-1120,1344,2520,1120,1680,105,-420,-1120,-420,112,210,-28,1
%N A324254 Signed version of the partition array A036039 (signed M_2 multinomial numbers).
%C A324254 The length of row n is p(n) = A000041(n), for n >= 1 (partition numbers).
%C A324254 The Abramowitz-Stegun order of partitions is used.
%C A324254 The triangle obtained by summing the terms belonging to like number of parts is A008275 (Stirling1).
%C A324254 This partition array T(n, k) and its partition row polynomials enter in the formula expressing the r-th power sums for the n-th elementary symmetric functions psigma(n, r) := Sum_{1 <= i1 < i2 < ... < in <= N} (x_{i1}*x_{i2}* ... *x_{in})^r in terms of the power sums of the first elementary function {ps(j*r)}_{j=1..n} with ps(i) := psigma(1, i) = Sum_{j = 1...N} (x_j)^i, for n >= 2 (for n = 1 it becomes an identity) and r >= 0. Note that psigma(n, 0) = binomial(N,n). The formula is given below.
%C A324254 The number N >= 1 of indeterminates is taken as fixed, and it is suppressed in the notations.
%C A324254 Note that only after the power sums have been replaced by the elementary symmetric functions (via the Girard-Waring formula) the result for the symmetric function psigma(n, r) becomes a function with integer coefficients (as guaranteed by the main theorem of symmetric functions).
%C A324254 The coefficients of partitions of 2*r with no more than r parts for psigma(2, r) in terms of the elementary symmetric functions {e_j} are given in array A308684.
%H A324254 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP?Res=150&amp;Page=831&amp;Submit=Go">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%F A324254 T(n, k) = (-1)^n*n!/Product_{j=1..n} (-1)^{a(n,k,j)}*j^a(n,k,j)*a(n,k,j)!, with the k-th partition of n >= 1 with m parts in Abramowitz-Stegun order written as Product_{j=1..n} j^a(n,k,j) with nonnegative integers a(n,k,j) satisfying Sum_{j=1..n} j*a(n,k,j) = n, for k = 1.. A000041(n), and the number of parts is Sum_{j=1..n} a(n,k,j) =: m(n,k). Hence the sign is (-1)^{n + m(n,k)}.
%F A324254 The formula for psigma(n, r), the r-th power sums of the n-th elementary symmetric functions in terms of the power sums {ps(j)}_{j=1..r*n}, is psigma(n, r) = (1/n!)*Sum_{k=1..p(n)} T(n,k) * Product_{j=1..n} ps(j*r)^{a(n,k,j)}, for n >= 1 and r >= 0, with the k-th partition of n as given in the T(n, k) formula.
%e A324254 The partition array T(n, k) begins:
%e A324254 n\k   1     2    3     4    5     6    7    8     9    10    11  12   13   14 15
%e A324254 --------------------------------------------------------------------------------
%e A324254 1:    1
%e A324254 2:   -1     1
%e A324254 3:    2    -3    1
%e A324254 4    -6     8    3    -6    1
%e A324254 5    24   -30  -20    20   15   -10    1
%e A324254 6  -120   144   90    40  -90  -120  -15   40    45   -15    1
%e A324254 7   720  -840 -504  -420  504   630  280  210  -210  -420  -105  70  105  -21  1
%e A324254 ...
%e A324254 n = 8: [-5040] [5760, 3360, 2688, 1260] [-3360, -4032, -3360, -1260, -1120] [1344, 2520, 1120, 1680, 105] [-420, -1120, -420] [112, 210] [-28] [1];
%e A324254 n = 9: [40320] [-45360, -25920, -20160, -18144] [25920, 30240, 24192, 11340, 9072, 15120, 2240] [-10080, -18144, -15120, -11340, -10080, -2520] [3024, 7560, 3360, 7560, 945] [-756, -2520, -1260] [168, 378] [-36] [1];
%e A324254 n = 10: [-362880] [403200, 226800, 172800, 151200, 72576] [-226800, -259200, -201600, -181440, -75600, -120960, -56700, -50400] [86400, 151200, 120960, 56700, 90720, 151200, 22400, 18900, 25200] [-25200, -60480, -50400, -56700, -50400, -25200, -945] [6048, 18900, 8400, 25200, 4725] [-1260, -5040, -3150] [240, 630] [-45] [1];
%e A324254 The brackets collect numbers belonging to the same number of parts m = m(n,k).
%e A324254 ...
%e A324254 --------------------------------------------------------------------------------
%e A324254 The first psigma(n, r) are: (the N indeterminates are suppressed)
%e A324254 psigma(1, r) = ps(1*r);
%e A324254 psigma(2, r) = (1/2)*(-ps(2*r) + ps(r)^2);
%e A324254 psigma(3, r) = (1/3!)*(2*ps(3*r) - 3*ps(1*r)*ps(2*r) + ps(1*r)^3);
%e A324254 psigma(4, r) = (1/4!)*(-6*ps(4*r) + 8*ps(1*r)*ps(3*r) + 3*ps(2*r)^2 - 6*ps(1*r)^2*ps(2*r) + ps(1*r)^4);
%e A324254 psigma(5, r) = (1/5!)*(24*ps(5*r) - 30*ps(1*r)*p(4*r) - 20*ps(2*r)^2*ps(3*r) + 20*ps(1*r)^2*ps(3*r) + 15*ps(1*r)*ps(2*r)^2 - - 10*ps(1*r)^3*ps(2*r) + 1*ps(1*r)^5);
%e A324254 psigma(6, r) = (1/6!)*(-120*ps(6*r) + 144*ps(1*r)*ps(5*r) + 90*ps(2*r)*ps(4*r) + 40*(ps(3*r))^2 - 90*ps(1*r)^2*ps(4*r) - 120*ps(1*r)*ps(2*r)*ps(3*r) - 15*ps(2*r)^3 + 40*(ps(1*r))^3*ps(3*r) + 45*ps(1*r)^2*ps(2*r)^2 - 15*ps(1*r)^4*ps(2*r) + ps(1*r)^6:
%e A324254 ...
%e A324254 --------------------------------------------------------------------------------
%Y A324254 Cf. A008275, A036039 (unsigned), A308684.
%K A324254 sign,tabf
%O A324254 1,4
%A A324254 _Wolfdieter Lang_, Jul 05 2019