Two-sided permutation statistics via symmetric functions
Given a permutation statistic $\operatorname {\mathrm {st}}$ , define its inverse statistic $\operatorname {\mathrm {ist}}$ by . We give a general approach, based on the theory of symmetric functions, for finding the joint distribution of $\operatorname {\mathrm {st}}_{1}$ and...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Cambridge University Press
2024-01-01
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| Series: | Forum of Mathematics, Sigma |
| Subjects: | |
| Online Access: | https://www.cambridge.org/core/product/identifier/S2050509424000896/type/journal_article |
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| Summary: | Given a permutation statistic
$\operatorname {\mathrm {st}}$
, define its inverse statistic
$\operatorname {\mathrm {ist}}$
by . We give a general approach, based on the theory of symmetric functions, for finding the joint distribution of
$\operatorname {\mathrm {st}}_{1}$
and
$\operatorname {\mathrm {ist}}_{2}$
whenever
$\operatorname {\mathrm {st}}_{1}$
and
$\operatorname {\mathrm {st}}_{2}$
are descent statistics: permutation statistics that depend only on the descent composition. We apply this method to a number of descent statistics, including the descent number, the peak number, the left peak number, the number of up-down runs and the major index. Perhaps surprisingly, in many cases the polynomial giving the joint distribution of
$\operatorname {\mathrm {st}}_{1}$
and
$\operatorname {\mathrm {ist}}_{2}$
can be expressed as a simple sum involving products of the polynomials giving the (individual) distributions of
$\operatorname {\mathrm {st}}_{1}$
and
$\operatorname {\mathrm {st}}_{2}$
. Our work leads to a rederivation of Stanley’s generating function for doubly alternating permutations, as well as several conjectures concerning real-rootedness and
$\gamma $
-positivity. |
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| ISSN: | 2050-5094 |