OFFSET
1,35
COMMENTS
The length of row n is A000041(n).
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).
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.
EXAMPLE
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;
...
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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
KEYWORD
sign,tabf
AUTHOR
Wolfdieter Lang, May 23 2019
STATUS
approved