A324247 Partition array giving in row n, for n >= 1, the coefficients of the Witt symmetric function w_n in terms of the elementary symmetric functions (using partitions in the Abramowitz-Stegun order).
1, -1, 0, 1, -1, 0, -1, 1, 0, -1, 0, 1, -1, -1, 1, 1, -1, 0, -1, 1, 1, 0, -1, -1, 0, 1, 1, -1, 0, 1, -1, -1, -1, 1, 2, 1, 1, -1, -3, -1, 1, 2, -1, 0, -1, 1, 1, 1, 0, -1, -2, -1, -1, -1, 1, 2, -2, 3, 0, -1, -3, 0, 1, 2, -1, 0, 1, -1, -1, -1, -1, 1, 2, 2, 1, 1, 2, 0, -1, -3, -3, -3, -2, -1, 1, 4, 2, 5, 1, -1, -5, -3, 1, 3, -1, 0, -1, 1, 1, 1, 1, 0, -1, -2, -2, -1, -1, -1, -1, -1, 1, 3, 2, 1, 2, 5, 1, 1, 1, -1, -3, -3, -5, -5, -3, 0, 1, 4, 2, 8, 2, -1, -5, -4, 1, 3, -1, 0
Offset: 1
Examples
The irregular triangle (partition array) begins: n\k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 ... ------------------------------------------------------------------------------ 1: 1 2: -1 0 3: 1 -1 0 4: -1 1 0 -1 0 5: 1 -1 -1 1 1 -1 0 6: -1 1 1 0 -1 -1 0 1 1 -1 0 7: 1 -1 -1 -1 1 2 1 1 -1 -3 -1 1 2 -1 0 8: -1 1 1 1 0 -1 -2 -1 -1 -1 1 2 -2 3 0 -1 -3 0 1 2 -1 0 ... n = 9: 1 -1 -1 -1 -1 1 2 2 1 1 2 0 -1 -3 -3 -3 -2 -1 1 4 2 5 1 -1 -5 -3 1 3 -1 0; n = 10: -1 1 1 1 1 0 -1 -2 -2 -1 -1 -1 -1 -1 1 3 2 1 2 5 1 1 1 -1 -3 -3 -5 -5 -3 0 1 4 2 8 2 -1 -5 -4 1 3 -1 0; ... --------------------------------------------------------------------------------- w_1 = e_1; w_2 = - e_2 + 0; w_3 = e_3 - e_1*e_2 + 0; w_4:= - e_4 + e_1*e_3 + 0 - (e_1)^2*e_2 + 0; w_5 = e_5 - e_1*e_4 - e_2*e_3 + (e_1)^2*e_3 + e_1*(e_2)^2 - (e_1)^3*e_2 + 0; w_6 = - e_6 + e_1*e_5 + e_2*e_4 + 0 - (e_1)^2*e_4 - e_1*e_2*e_3 + 0 + (e_1)^3*e_3 + (e_1)^2*(e_2)^2 - (e_1)^4*e_2 + 0; ... ---------------------------------------------------------------------------------
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]
- H J. Borger, Witt vectors, semirings, and total positivity, arXiv:1310.3013 [math.CO], 2015, Section 4.5., pp. 295-296 [with theta -> w, and the n = 1..6 results on p. 295]
- SAGE, Witt symmetric functions
Formula
w_n is given by the recurrence given in the comment above via the power sum symmetric functions {p_i} expressed in terms of the elementary symmetric functions {e_i}.
T(n, k) gives the coefficient of (e_1)^{a(k,1)}* ... *(e_n)^{a(k,n)} for w_n, corresponding to the k-th partition of n in Abramowitz-Stegun order, written as 1^{a(k,1)}* ... *n^{a(k,n)}, with nonnegative integers a(k,j) satisfying Sum_{j=1..n} j*a(k,j) = n, and the number of parts is Sum_{j=1..n} a(k,j) =: m.
Comments