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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).

%I A324247 #21 Dec 15 2024 15:46:44
%S A324247 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,
%T A324247 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,
%U A324247 -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
%N 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).
%C A324247 The length of row n is A000041(n).
%C A324247 The (one part) Witt symmetric function w_n is defined in the links below (one can add w_0 = 1). It can be expressed in terms of the elementary symmetric functions {e_i}_{i=1..n} by using first a recurrence to express w_n in terms of the power sum symmetric functions p_n = Sum_{1>=1} x_i^n, for the indeterminates {x_i}, by w_n = (1/n)*(p_n - Sum_{d|n, 1 <= d < n} d*(w_d)^{n/d}), n >= 2, with w_1 = p_1 = e_1. (See the array A324253). The p_n can then be expressed in terms of {e_i}_{i=1..n} by the Newton recurrence or its solution, the Girard-Waring formula (see A115131, row n, with partitions in the Abramowitz-Stegun order).
%C A324247 A relation between {w_n}_{n>=1}, {e_i}_{i>=0}, with e_0 = 1, and the indeterminates {x_i}_{i>=1} is: Product_{n>=0}(1 - w_n*t^n) = Sum_{i>=0} e_i*(-t)^i = Product_{j>=1} (1 - x_j*t). See the links.
%C A324247 If only N indeterminates {x_i}_{i=1..N} are considered all coefficients corresponding to partitions with at least one part > N are set to 0 (in addition to the ones given in the sequence).
%H A324247 M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP?Res=150&amp;Page=821">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy]
%H A324247 H J. Borger, <a href="https://arxiv.org/abs/1310.3013">Witt vectors, semirings, and total positivity</a>, arXiv:1310.3013 [math.CO], 2015, Section 4.5., pp. 295-296 [with theta -> w, and the n = 1..6 results on p. 295]
%H A324247 SAGE, <a href="https://www.math.sciences.univ-nantes.fr/~sorger/chow/html/en/reference/combinat/sage/combinat/sf/witt.html">Witt symmetric functions</a>
%F A324247 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}.
%F A324247 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.
%e A324247 The irregular triangle (partition array) begins:
%e A324247 n\k  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 21 22 ...
%e A324247 ------------------------------------------------------------------------------
%e A324247 1:   1
%e A324247 2:  -1  0
%e A324247 3:   1 -1  0
%e A324247 4:  -1  1  0 -1  0
%e A324247 5:   1 -1 -1  1  1 -1  0
%e A324247 6:  -1  1  1  0 -1 -1  0  1  1 -1  0
%e A324247 7:   1 -1 -1 -1  1  2  1  1 -1 -3 -1  1  2 -1  0
%e A324247 8:  -1  1  1  1  0 -1 -2 -1 -1 -1  1  2 -2  3  0 -1 -3  0  1  2 -1  0
%e A324247 ...
%e A324247 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;
%e A324247 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;
%e A324247 ...
%e A324247 ---------------------------------------------------------------------------------
%e A324247 w_1 = e_1;
%e A324247 w_2 = - e_2 + 0;
%e A324247 w_3 = e_3 - e_1*e_2 + 0;
%e A324247 w_4:= - e_4 + e_1*e_3 + 0 - (e_1)^2*e_2 + 0;
%e A324247 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;
%e A324247 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;
%e A324247 ...
%e A324247 ---------------------------------------------------------------------------------
%Y A324247 Cf. A000041, A115131 (Waring numbers), A324253 (with power sums).
%K A324247 sign,tabf
%O A324247 1,35
%A A324247 _Wolfdieter Lang_, May 23 2019