# Counting descents, rises, and levels, with prescribed first element, in words

Sergey Kitaev, Toufik Mansour, Jeffrey Remmel

Research output: Contribution to journalArticle

5 Citations (Scopus)

### Abstract

Recently, Kitaev and Remmel  re¯ned the well-known permutation statistic \descent" by ¯xing parity of one of the descent's numbers. Results in  were extended and generalized in several ways in [7, 10, 11, 12]. In this paper, we shall ¯x a set partition of the natural numbers N, (N1; : : : ;Ns), and we study the distribution of descents, levels, and rises according to whether the ¯rst letter of the descent, rise, or level lies in Ni over the set of words over the alphabet [k] = f1; : : : ; kg. In particular, we re¯ne and generalize some of the results in .
Original language English 22 Discrete Mathematics and Theoretical Computer Science 10 3 Published - 2008

### Fingerprint

Descent
Counting
Statistics
Permutation Statistics
Set Partition
Natural number
Parity
Generalise

### Keywords

• distribution of descents
• rises
• levels
• permutation statistics

### Cite this

@article{9d457bf3edbb4b7789d2ceda6de72f76,
title = "Counting descents, rises, and levels, with prescribed first element, in words",
abstract = "Recently, Kitaev and Remmel  re¯ned the well-known permutation statistic \descent{"} by ¯xing parity of one of the descent's numbers. Results in  were extended and generalized in several ways in [7, 10, 11, 12]. In this paper, we shall ¯x a set partition of the natural numbers N, (N1; : : : ;Ns), and we study the distribution of descents, levels, and rises according to whether the ¯rst letter of the descent, rise, or level lies in Ni over the set of words over the alphabet [k] = f1; : : : ; kg. In particular, we re¯ne and generalize some of the results in .",
keywords = "distribution of descents, rises, levels, permutation statistics",
author = "Sergey Kitaev and Toufik Mansour and Jeffrey Remmel",
year = "2008",
language = "English",
volume = "10",
journal = "Discrete Mathematics and Theoretical Computer Science",
issn = "1365-8050",
publisher = "Maison de l'informatique et des mathematiques discretes",
number = "3",

}

Counting descents, rises, and levels, with prescribed first element, in words. / Kitaev, Sergey; Mansour, Toufik; Remmel, Jeffrey.

In: Discrete Mathematics and Theoretical Computer Science, Vol. 10, No. 3, 2008.

Research output: Contribution to journalArticle

TY - JOUR

T1 - Counting descents, rises, and levels, with prescribed first element, in words

AU - Kitaev, Sergey

AU - Mansour, Toufik

AU - Remmel, Jeffrey

PY - 2008

Y1 - 2008

N2 - Recently, Kitaev and Remmel  re¯ned the well-known permutation statistic \descent" by ¯xing parity of one of the descent's numbers. Results in  were extended and generalized in several ways in [7, 10, 11, 12]. In this paper, we shall ¯x a set partition of the natural numbers N, (N1; : : : ;Ns), and we study the distribution of descents, levels, and rises according to whether the ¯rst letter of the descent, rise, or level lies in Ni over the set of words over the alphabet [k] = f1; : : : ; kg. In particular, we re¯ne and generalize some of the results in .

AB - Recently, Kitaev and Remmel  re¯ned the well-known permutation statistic \descent" by ¯xing parity of one of the descent's numbers. Results in  were extended and generalized in several ways in [7, 10, 11, 12]. In this paper, we shall ¯x a set partition of the natural numbers N, (N1; : : : ;Ns), and we study the distribution of descents, levels, and rises according to whether the ¯rst letter of the descent, rise, or level lies in Ni over the set of words over the alphabet [k] = f1; : : : ; kg. In particular, we re¯ne and generalize some of the results in .

KW - distribution of descents

KW - rises

KW - levels

KW - permutation statistics

UR - http://www.dmtcs.org/dmtcs-ojs/index.php/dmtcs/article/view/680

UR - https://personal.cis.strath.ac.uk/sergey.kitaev/index_files/Papers/deslevris-words.pdf

M3 - Article

VL - 10

JO - Discrete Mathematics and Theoretical Computer Science

JF - Discrete Mathematics and Theoretical Computer Science

SN - 1365-8050

IS - 3

ER -