Enumeration of 3-letter patterns in compositions

Let A be any set of positive integers and n a positive integer. A composition of n with parts in A is an ordered collection of one or more elements in A whose sum is n. We derive generating functions for the number of compositions of n with m parts in A that have r occurrences of 3-letter patterns formed by two (adjacent) instances of levels, rises and drops. We also derive asymptotics for the number of compositions of n that avoid a given pattern. Finally, we obtain the generating function for the number of k-ary words of length m which contain a prescribed number of occurrences of a given pattern as a special case of our results.Comment: 20 pages, 1 figure; accepted for the Proceedings of the 2005 Integer Conferenc

Part-products of SS-restricted integer compositions

If $S$ is a cofinite set of positive integers, an "$S$-restricted composition of $n$" is a sequence of elements of $S$, denoted $\vec{\lambda}=(\lambda_1,\lambda_2,...)$, whose sum is $n$. For uniform random $S$-restricted compositions, the random variable ${\bf B}(\vec{\lambda})=\prod_i \lambda_i$ is asymptotically lognormal. The proof is based upon a combinatorial technique for decomposing a composition into a sequence of smaller compositions.Comment: 18 page

Horizontal runs in domino tilings

We discuss tilings of a grid (of size n × 2) with dominoes of size 2 × 1. Parameters that might be called "longest run" are investigated, in terms of generating functions and also asymptotically. Extensions are also mentioned

Five Guidelines for Partition Analysis with Applications to Lecture Hall-type Theorems

Five simple guidelines are proposed to compute the generating function for the nonnegative integer solutions of a system of linear inequalities. In contrast to other approaches, the emphasis is on deriving recurrences. We show how to use the guidelines strategically to solve some nontrivial enumeration problems in the theory of partitions and compositions. This includes a strikingly different approach to lecture hall-type theorems, with new $q$-series identities arising in the process. For completeness, we prove that the guidelines suffice to find the generating function for any system of homogeneous linear inequalities with integer coefficients. The guidelines can be viewed as a simplification of MacMahon's partition analysis with ideas from matrix techiniques, Elliott reduction, and ``adding a slice''

Tight Markov chains and random compositions

For an ergodic Markov chain $\{X(t)\}$ on $\Bbb N$, with a stationary distribution $\pi$, let $T_n>0$ denote a hitting time for $[n]^c$, and let $X_n=X(T_n)$. Around 2005 Guy Louchard popularized a conjecture that, for $n\to \infty$, $T_n$ is almost Geometric($p$), $p=\pi([n]^c)$, $X_n$ is almost stationarily distributed on $[n]^c$, and that $X_n$ and $T_n$ are almost independent, if $p(n):=\sup_ip(i,[n]^c)\to 0$ exponentially fast. For the chains with $p(n) \to 0$ however slowly, and with $\sup_{i,j}\,\|p(i,\cdot)-p(j,\cdot)\|_{TV}<1$, we show that Louchard's conjecture is indeed true even for the hits of an arbitrary $S_n\subset\Bbb N$ with $\pi(S_n)\to 0$. More precisely, a sequence of $k$ consecutive hit locations paired with the time elapsed since a previous hit (for the first hit, since the starting moment) is approximated, within a total variation distance of order $k\,\sup_ip(i,S_n)$, by a $k$-long sequence of independent copies of $(\ell_n,t_n)$, where $\ell_n= \text{Geometric}\,(\pi(S_n))$, $t_n$ is distributed stationarily on $S_n$, and $\ell_n$ is independent of $t_n$. The two conditions are easily met by the Markov chains that arose in Louchard's studies as likely sharp approximations of two random compositions of a large integer $\nu$, a column-convex animal (cca) composition and a Carlitz (C) composition. We show that this approximation is indeed very sharp for most of the parts of the random compositions. Combining the two approximations in a tandem, we are able to determine the limiting distributions of $\mu=o(\ln\nu)$ and $\mu=o(\nu^{1/2})$ largest parts of the random cca composition and the random C-composition, respectively. (Submitted to Annals of Probability in August, 2009.

The distribution of ascents of size d or more in compositions

Combinatoric

Locally Restricted Compositions IV. Nearly Free Large Parts and Gap-Freeness

We define the notion of $t$-free for locally restricted compositions, which means roughly that if such a composition contains a part $c_i$ and nearby parts are at least $t$ smaller, then $c_i$ can be replaced by any larger part. Two well-known examples are Carlitz and alternating compositions. We show that large parts have asymptotically geometric distributions. This leads to asymptotically independent Poisson variables for numbers of various large parts. Based on this we obtain asymptotic formulas for the probability of being gap free and for the expected values of the largest part and number distinct parts, all accurate to $o(1)$