Articles written in Proceedings – Mathematical Sciences

    • Enumerating Set Partitions According to the Number of Descents of Size 𝑑 or more

      Toufik Mansour Mark Shattuck Chunwei Song

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      Let $P(n,k)$ denote the set of partitions of $\{1,2,\ldots,n\}$ having exactly 𝑘 blocks. In this paper, we find the generating function which counts the members of $P(n,k)$ according to the number of descents of size 𝑑 or more, where $d\geq 1$ is fixed. An explicit expression in terms of Stirling numbers of the second kind may be given for the total number of such descents in all the members of $P(n,k)$. We also compute the generating function for the statistics recording the number of ascents of size 𝑑 or more and show that it has the same distribution on $P(n,k)$ as the prior statistics for descents when $d\geq 2$, by both algebraic and combinatorial arguments.

    • A Statistic on 𝑛-Color Compositions and Related Sequences

      Toufik Mansour Mark Shattuck

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      A composition of a positive integer in which a part of size 𝑛 may be assigned one of 𝑛 colors is called an 𝑛-color composition. Let $a_m$ denote the number of 𝑛-color compositions of the integer 𝑚. It is known that $a_m = F_{2m}$ for all $m \geq 1$, where $F_m$ denotes the Fibonacci number defined by $F_m = F_{m-1}+F_{m-2}$ if $m\geq 2$, with $F_0=0$ and $F_1=1$. A statistic is studied on the set of 𝑛-color compositions of 𝑚 thus providing a polynomial generalization of the sequence $F_{2m}$. The statistic may be described, equivalently, in terms of statistics on linear tilings and lattice paths. The restriction to the set of 𝑛-color compositions having a prescribed number of parts is considered and an explicit formula for the distribution is derived. We also provide 𝑞-generalizations of relations between $a_m$ and the number of self-inverse 𝑛-compositions of $2m+1$ or $2m$. Finally, we consider a more general recurrence than that satisfied by the numbers $a_m$ and note some particular cases.

    • Counting rises and levels in $r$-color compositions


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      An $r$-color composition of a positive integer $n$ is a sequence of positiveintegers, called parts, summing to n in which each part of size $r$ is assigned one of $r$ possible colors. In this paper, we address the problem of counting the $r$-color compositions having a prescribed number of rises. Formulas for the relevant generating functions are computed which count the compositions in question according to a certain statistic. Furthermore, we find explicit formulas for the total number of rises within all of the $r$-color compositions of $n$ having a fixed number of parts. A similar treatment is given for the problem of counting the number of levels and a further generalization in terms of rises of a particular type is discussed.

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