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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).
(history;
published version)
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#17 by Bruno Berselli at Thu Aug 29 11:43:30 EDT 2019
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#16 by Joerg Arndt at Thu Aug 29 11:19:59 EDT 2019
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#15 by Omar E. Pol at Thu Aug 29 11:15:20 EDT 2019
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#14 by Omar E. Pol at Thu Aug 29 11:14:20 EDT 2019
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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 partitonspartitions in the Abramowitz-Stegun order).
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| STATUS
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approved
editing
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Discussion
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| Thu Aug 29
| 11:15
| Omar E. Pol: Typo corrected. I think that Data section is too long
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#13 by Bruno Berselli at Tue Jun 25 03:23:39 EDT 2019
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#12 by Michel Marcus at Tue Jun 25 03:22:58 EDT 2019
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#11 by Vaclav Kotesovec at Tue Jun 25 03:13:43 EDT 2019
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#10 by Vaclav Kotesovec at Tue Jun 25 03:13:36 EDT 2019
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| CROSSREFS
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Cf. A000041, A11513A115131 (Waring numbers), A324253 (with power sums).
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approved
editing
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#9 by Peter Luschny at Mon Jun 24 17:49:59 EDT 2019
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#8 by Jon E. Schoenfield at Wed Jun 05 21:05:19 EDT 2019
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