A positional statistic for 1324-avoiding permutations
We consider the class $S_n(1324)$ of permutations of size $n$ that avoid the pattern 1324 and examine the subset $S_n^{a\prec n}(1324)$ of elements for which $a\prec n\prec [a-1]$, $a\ge 1$. This notation means that, when written in one line notation, such a permutation must have $a$ to the left of...
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| Published in: | Discrete mathematics and theoretical computer science Vol. 26:1, Permutation...; no. Special issues |
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| Main Authors: | , , |
| Format: | Journal Article |
| Language: | English |
| Published: |
Discrete Mathematics & Theoretical Computer Science
04.11.2024
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| Subjects: | |
| ISSN: | 1365-8050, 1365-8050 |
| Online Access: | Get full text |
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| Summary: | We consider the class $S_n(1324)$ of permutations of size $n$ that avoid the
pattern 1324 and examine the subset $S_n^{a\prec n}(1324)$ of elements for
which $a\prec n\prec [a-1]$, $a\ge 1$. This notation means that, when written
in one line notation, such a permutation must have $a$ to the left of $n$, and
the elements of $\{1,\dots,a-1\}$ must all be to the right of $n$. For $n\ge
2$, we establish a connection between the subset of permutations in
$S_n^{1\prec n}(1324)$ having the 1 adjacent to the $n$ (called primitives),
and the set of 1324-avoiding dominoes with $n-2$ points. For $a\in\{1,2\}$, we
introduce constructive algorithms and give formulas for the enumeration of
$S_n^{a\prec n}(1324)$ by the position of $a$ relative to the position of $n$.
For $a\ge 3$, we formulate some conjectures for the corresponding generating
functions. |
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| ISSN: | 1365-8050 1365-8050 |
| DOI: | 10.46298/dmtcs.12629 |