# Equidistribution of descents, adjacent pairs, and place-value pairs on permutations

Emeric Deutsch, Sergey Kitaev, Jeffrey Remmel

Research output: Contribution to journalArticle

### Abstract

An \$(X,Y)\$-descent in a permutation is a pair of adjacent elements such that the first element is from \$X\$, the second element is from \$Y\$, and the first element is greater than the second one. An \$(X,Y)\$-adjacency in a permutation is a pair of adjacent elements such that the first one is from \$X\$ and the second one is from \$Y\$. An \$(X,Y)\$-place-value pair in a permutation is an element \$y\$ in position \$x\$, such that \$y\$ is in \$Y\$ and \$x\$ is in \$X\$. It turns out, that for certain choices of \$X\$ and \$Y\$ some of the three statistics above become equidistributed. Moreover, it is easy to derive the distribution formula for \$(X,Y)\$-place-value pairs thus providing distribution for other statistics under consideration too. This generalizes some results in the literature. As a result of our considerations, we get combinatorial proofs of several remarkable identities. We also conjecture existence of a bijection between two objects in question preserving a certain statistic.
Original language English 09.5.1 19 Journal of Integer Sequences 12 5 Published - 2009

Place value
Equidistribution
Descent
Permutation
Statistics
Bijection
Statistic
Generalise

• permutations
• bijection

### Cite this

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title = "Equidistribution of descents, adjacent pairs, and place-value pairs on permutations",
abstract = "An \$(X,Y)\$-descent in a permutation is a pair of adjacent elements such that the first element is from \$X\$, the second element is from \$Y\$, and the first element is greater than the second one. An \$(X,Y)\$-adjacency in a permutation is a pair of adjacent elements such that the first one is from \$X\$ and the second one is from \$Y\$. An \$(X,Y)\$-place-value pair in a permutation is an element \$y\$ in position \$x\$, such that \$y\$ is in \$Y\$ and \$x\$ is in \$X\$. It turns out, that for certain choices of \$X\$ and \$Y\$ some of the three statistics above become equidistributed. Moreover, it is easy to derive the distribution formula for \$(X,Y)\$-place-value pairs thus providing distribution for other statistics under consideration too. This generalizes some results in the literature. As a result of our considerations, we get combinatorial proofs of several remarkable identities. We also conjecture existence of a bijection between two objects in question preserving a certain statistic.",
keywords = "adjacent pairs, permutations, bijection",
author = "Emeric Deutsch and Sergey Kitaev and Jeffrey Remmel",
year = "2009",
language = "English",
volume = "12",
journal = "Journal of Integer Sequences",
issn = "1530-7638",
number = "5",

}

Equidistribution of descents, adjacent pairs, and place-value pairs on permutations. / Deutsch, Emeric; Kitaev, Sergey; Remmel, Jeffrey.

In: Journal of Integer Sequences, Vol. 12, No. 5, 09.5.1, 2009.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Equidistribution of descents, adjacent pairs, and place-value pairs on permutations

AU - Deutsch, Emeric

AU - Kitaev, Sergey

AU - Remmel, Jeffrey

PY - 2009

Y1 - 2009

N2 - An \$(X,Y)\$-descent in a permutation is a pair of adjacent elements such that the first element is from \$X\$, the second element is from \$Y\$, and the first element is greater than the second one. An \$(X,Y)\$-adjacency in a permutation is a pair of adjacent elements such that the first one is from \$X\$ and the second one is from \$Y\$. An \$(X,Y)\$-place-value pair in a permutation is an element \$y\$ in position \$x\$, such that \$y\$ is in \$Y\$ and \$x\$ is in \$X\$. It turns out, that for certain choices of \$X\$ and \$Y\$ some of the three statistics above become equidistributed. Moreover, it is easy to derive the distribution formula for \$(X,Y)\$-place-value pairs thus providing distribution for other statistics under consideration too. This generalizes some results in the literature. As a result of our considerations, we get combinatorial proofs of several remarkable identities. We also conjecture existence of a bijection between two objects in question preserving a certain statistic.

AB - An \$(X,Y)\$-descent in a permutation is a pair of adjacent elements such that the first element is from \$X\$, the second element is from \$Y\$, and the first element is greater than the second one. An \$(X,Y)\$-adjacency in a permutation is a pair of adjacent elements such that the first one is from \$X\$ and the second one is from \$Y\$. An \$(X,Y)\$-place-value pair in a permutation is an element \$y\$ in position \$x\$, such that \$y\$ is in \$Y\$ and \$x\$ is in \$X\$. It turns out, that for certain choices of \$X\$ and \$Y\$ some of the three statistics above become equidistributed. Moreover, it is easy to derive the distribution formula for \$(X,Y)\$-place-value pairs thus providing distribution for other statistics under consideration too. This generalizes some results in the literature. As a result of our considerations, we get combinatorial proofs of several remarkable identities. We also conjecture existence of a bijection between two objects in question preserving a certain statistic.

KW - permutations

KW - bijection

UR - https://cs.uwaterloo.ca/journals/JIS/VOL12/Kitaev/kitaev4.pdf

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VL - 12

JO - Journal of Integer Sequences

JF - Journal of Integer Sequences

SN - 1530-7638

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ER -