# Frame patterns in n-cycles

Miles Jones, Sergey Kitaev, Jeffrey Remmel

Research output: Contribution to journalArticle

2 Citations (Scopus)

### Abstract

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 $n$.
Original language English 1197-1215 19 Discrete Mathematics 338 7 28 Feb 2015 https://doi.org/10.1016/j.disc.2015.01.033 Published - 6 Jul 2015

### Fingerprint

Statistics
Polynomials
Cycle
Q-analogue
Permutation
Charge
Dyck Paths
Clockwise
Binomial coefficient
Rearrangement
Symmetric group
Generating Function
Statistic
Immediately
Linear Combination
Partition
Path
Polynomial
Coefficient

### Keywords

• frame patterns
• N-cycle
• linear combinatorics

### Cite this

Jones, Miles ; Kitaev, Sergey ; Remmel, Jeffrey. / Frame patterns in n-cycles. In: Discrete Mathematics. 2015 ; Vol. 338, No. 7. pp. 1197-1215.
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Frame patterns in n-cycles. / Jones, Miles; Kitaev, Sergey; Remmel, Jeffrey.

In: Discrete Mathematics, Vol. 338, No. 7, 06.07.2015, p. 1197-1215.

Research output: Contribution to journalArticle

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 $n$.

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KW - N-cycle

KW - linear combinatorics

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