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A324247
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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).
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1
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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
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OFFSET
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1,35
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COMMENTS
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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).
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.
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).
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LINKS
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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]
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FORMULA
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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.
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EXAMPLE
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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 ...
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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;
...
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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;
...
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CROSSREFS
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KEYWORD
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sign,tabf
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AUTHOR
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STATUS
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approved
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