### Abstract

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

Pages (from-to) | 1197-1215 |

Number of pages | 19 |

Journal | Discrete Mathematics |

Volume | 338 |

Issue number | 7 |

Early online date | 28 Feb 2015 |

DOIs | |

Publication status | Published - 6 Jul 2015 |

### Fingerprint

### Keywords

- frame patterns
- N-cycle
- linear combinatorics

### Cite this

*Discrete Mathematics*,

*338*(7), 1197-1215. https://doi.org/10.1016/j.disc.2015.01.033

}

*Discrete Mathematics*, vol. 338, no. 7, pp. 1197-1215. https://doi.org/10.1016/j.disc.2015.01.033

**Frame patterns in n-cycles.** / Jones, Miles; Kitaev, Sergey; Remmel, Jeffrey.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Frame patterns in n-cycles

AU - Jones, Miles

AU - Kitaev, Sergey

AU - Remmel, Jeffrey

PY - 2015/7/6

Y1 - 2015/7/6

N2 - In this paper, we study the distribution of the number occurrences of the simplest frame pattern, called the $\mu$ pattern, in $n$-cycles. Given an $n$-cycle $C$, we say that a pair $\langle i,j \rangle$ matches the $\mu$ pattern if $i < j$ and as we traverse around $C$ in a clockwise direction starting at $i$ and ending at $j$, we never encounter a $k$ with $i < k < j$. We say that $ \lan i,j \ran$ is a nontrivial $\mu$-match if $i+1 < j$. We say that an $n$-cycle $C$ is incontractible if there is no $i$ such that $i+1$ immediately follows $i$ in $C$. We show that number of incontractible $n$-cycles in the symmetric group $S_n$ is $D_{n-1}$ where $D_n$ is the number of derangements in $S_n$. We show that number of $n$-cycles in $S_n$ with exactly $k$ $\mu$-matches can be expressed as a linear combination of binomial coefficients of the form $\binom{n-1}{i}$ where $i \leq 2k+1$. We also show that the generating function $NTI_{n,\mu}(q)$ of $q$ raised to the number of nontrivial $\mu$-matches in $C$ over all incontractible $n$-cycles in $S_n$ is a new $q$-analogue of $D_{n-1}$ which is different from the $q$-analogues of the derangement numbers that have been studied by Garsia and Remmel and by Wachs. We will show that there is a rather surprising connection between the charge statistic on permutations due to Lascoux and Sch\"uzenberger and our polynomials in that the coefficient of the smallest power of $q$ in $NTI_{2k+1,\mu}(q)$ is the number of permutations in $S_{2k+1}$ whose charge path is a Dyck path. Finally, we show that $NTI_{n,\mu}(q)|_{q^{\binom{n-1}{2} -k}}$ and $NT_{n,\mu}(q)|_{q^{\binom{n-1}{2} -k}}$ are the number of partitions of $k$ for sufficiently large $n$.

AB - In this paper, we study the distribution of the number occurrences of the simplest frame pattern, called the $\mu$ pattern, in $n$-cycles. Given an $n$-cycle $C$, we say that a pair $\langle i,j \rangle$ matches the $\mu$ pattern if $i < j$ and as we traverse around $C$ in a clockwise direction starting at $i$ and ending at $j$, we never encounter a $k$ with $i < k < j$. We say that $ \lan i,j \ran$ is a nontrivial $\mu$-match if $i+1 < j$. We say that an $n$-cycle $C$ is incontractible if there is no $i$ such that $i+1$ immediately follows $i$ in $C$. We show that number of incontractible $n$-cycles in the symmetric group $S_n$ is $D_{n-1}$ where $D_n$ is the number of derangements in $S_n$. We show that number of $n$-cycles in $S_n$ with exactly $k$ $\mu$-matches can be expressed as a linear combination of binomial coefficients of the form $\binom{n-1}{i}$ where $i \leq 2k+1$. We also show that the generating function $NTI_{n,\mu}(q)$ of $q$ raised to the number of nontrivial $\mu$-matches in $C$ over all incontractible $n$-cycles in $S_n$ is a new $q$-analogue of $D_{n-1}$ which is different from the $q$-analogues of the derangement numbers that have been studied by Garsia and Remmel and by Wachs. We will show that there is a rather surprising connection between the charge statistic on permutations due to Lascoux and Sch\"uzenberger and our polynomials in that the coefficient of the smallest power of $q$ in $NTI_{2k+1,\mu}(q)$ is the number of permutations in $S_{2k+1}$ whose charge path is a Dyck path. Finally, we show that $NTI_{n,\mu}(q)|_{q^{\binom{n-1}{2} -k}}$ and $NT_{n,\mu}(q)|_{q^{\binom{n-1}{2} -k}}$ are the number of partitions of $k$ for sufficiently large $n$.

KW - frame patterns

KW - N-cycle

KW - linear combinatorics

UR - http://www.sciencedirect.com/science/journal/0012365X

UR - http://arxiv.org/abs/1311.3332

U2 - 10.1016/j.disc.2015.01.033

DO - 10.1016/j.disc.2015.01.033

M3 - Article

VL - 338

SP - 1197

EP - 1215

JO - Discrete Mathematics

JF - Discrete Mathematics

SN - 0012-365X

IS - 7

ER -