### Abstract

Original language | English |
---|---|

Article number | 09.5.1 |

Number of pages | 19 |

Journal | Journal of Integer Sequences |

Volume | 12 |

Issue number | 5 |

Publication status | Published - 2009 |

### Fingerprint

### Keywords

- adjacent pairs
- permutations
- bijection

### Cite this

*Journal of Integer Sequences*,

*12*(5), [09.5.1].

}

*Journal of Integer Sequences*, vol. 12, no. 5, 09.5.1.

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

Research output: Contribution to journal › Article

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 - adjacent pairs

KW - permutations

KW - bijection

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

M3 - Article

VL - 12

JO - Journal of Integer Sequences

JF - Journal of Integer Sequences

SN - 1530-7638

IS - 5

M1 - 09.5.1

ER -