A324254 Signed version of the partition array A036039 (signed M_2 multinomial numbers).
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, -1120, -420, 112, 210, -28, 1
Offset: 1
Examples
The partition array T(n, k) begins: n\k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 -------------------------------------------------------------------------------- 1: 1 2: -1 1 3: 2 -3 1 4 -6 8 3 -6 1 5 24 -30 -20 20 15 -10 1 6 -120 144 90 40 -90 -120 -15 40 45 -15 1 7 720 -840 -504 -420 504 630 280 210 -210 -420 -105 70 105 -21 1 ... n = 8: [-5040] [5760, 3360, 2688, 1260] [-3360, -4032, -3360, -1260, -1120] [1344, 2520, 1120, 1680, 105] [-420, -1120, -420] [112, 210] [-28] [1]; 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]; 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]; The brackets collect numbers belonging to the same number of parts m = m(n,k). ... -------------------------------------------------------------------------------- The first psigma(n, r) are: (the N indeterminates are suppressed) psigma(1, r) = ps(1*r); psigma(2, r) = (1/2)*(-ps(2*r) + ps(r)^2); psigma(3, r) = (1/3!)*(2*ps(3*r) - 3*ps(1*r)*ps(2*r) + ps(1*r)^3); 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); 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); 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: ... --------------------------------------------------------------------------------
Links
- 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].
Formula
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)}.
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.
Comments