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Register a new userEnumeration of $(k,2)$-noncrossing partitions
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A set partition is said to be $(k,d)$-noncrossing if it avoids the pattern $12... k12... d$. We find an explicit formula for the ordinary generating function of the number of $(k,d)$-noncrossing partitions of ${1,2,...,n}$ when $d=1,2$.
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For each skew shape we define a nonhomogeneous symmetric function, generalizing a construction of Pak and Postnikov. In two special cases, we show that the coefficients of this function when expanded in the complete homogeneous basis are given in terms of the (reduced) type of $k$-divisible noncrossing partitions. Our work extends Haimans notion of a parking function symmetric function.
In this paper we compute the generating function of modular, $k$-noncrossing diagrams. A $k$-noncrossing diagram is called modular if it does not contains any isolated arcs and any arc has length at least four. Modular diagrams represent the deformation retracts of RNA pseudoknot structures cite{Stadler:99,Reidys:07pseu,Reidys:07lego} and their properties reflect basic features of these bio-molecules. The particular case of modular noncrossing diagrams has been extensively studied cite{Waterman:78b, Waterman:79,Waterman:93, Schuster:98}. Let ${sf Q}_k(n)$ denote the number of modular $k$-noncrossing diagrams over $n$ vertices. We derive exact enumeration results as well as the asymptotic formula ${sf Q}_k(n)sim c_k n^{-(k-1)^2-frac{k-1}{2}}gamma_{k}^{-n}$ for $k=3,..., 9$ and derive a new proof of the formula ${sf Q}_2(n)sim 1.4848, n^{-3/2},1.8489^{-n}$ cite{Schuster:98}.
In this paper we study $k$-noncrossing matchings. A $k$-noncrossing matching is a labeled graph with vertex set ${1,...,2n}$ arranged in increasing order in a horizontal line and vertex-degree 1. The $n$ arcs are drawn in the upper halfplane subject to the condition that there exist no $k$ arcs that mutually intersect. We derive: (a) for arbitrary $k$, an asymptotic approximation of the exponential generating function of $k$-noncrossing matchings $F_k(z)$. (b) the asymptotic formula for the number of $k$-noncrossing matchings $f_{k}(n) sim c_k n^{-((k-1)^2+(k-1)/2)} (2(k-1))^{2n}$ for some $c_k>0$.
We consider $m$-divisible non-crossing partitions of ${1,2,ldots,mn}$ with the property that for some $tleq n$ no block contains more than one of the first $t$ integers. We give a closed formula for the number of multi-chains of such non-crossing partitions with prescribed number of blocks. Building on this result, we compute Chapotons $M$-triangle in this setting and conjecture a combinatorial interpretation for the $H$-triangle. This conjecture is proved for $m=1$.
Let $mge 2$ be a fixed positive integer. Suppose that $m^j leq n< m^{j+1}$ is a positive integer for some $jge 0$. Denote $b_{m}(n)$ the number of $m$-ary partitions of $n$, where each part of the partition is a power of $m$. In this paper, we show that $b_m(n)$ can be represented as a $j$-fold summation by constructing a one-to-one correspondence between the $m$-ary partitions and a special class of integer sequences rely only on the base $m$ representation of $n$. It directly reduces to Andrews, Fraenkel and Sellers characterization of the values $b_{m}(mn)$ modulo $m$. Moreover, denote $c_{m}(n)$ the number of $m$-ary partitions of $n$ without gaps, wherein if $m^i$ is the largest part, then $m^k$ for each $0leq k<i$ also appears as a part. We also obtain an enumeration formula for $c_m(n)$ which leads to an alternative representation for the congruences of $c_m(mn)$ due to Andrews, Fraenkel, and Sellers.
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