Figure 73 - from "Generating functions for composition/word statistics"

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Figure 7.6: A fixed point of Equation 7.2.1

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We say that o is an up-down permutation if Notice that there are three analogues for des and maj, respectively. It turns

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Evan Fuller

Generating functions for composition/word statistics

Abstract: In Chapter 6, we apply an alternate method to find the generating functions for certain classes of alternating words on alphabet [Special characters omitted.] In addition, we use the results of previous chapters to find generating functions for statistics defined by [Special ...

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Abstract: A composition $\pi = \pi_1 \pi_2 \cdots \pi_m$ of a positive integer $n$ is an ordered collection of one or more positive integers whose sum is $n$. The number of summands, namely $m$, is called the number of parts of $\pi$. Using linear algebra, we determine formulas for generating functions that count compositions of $n$ with $m$ parts, according to the number of occurrences of the subword pattern $\tau$, and according to the sum, over all occurrences of $\tau$, of the first integers in their respective occurrences, where $\tau$ is any pattern of length three with exactly 2 distinct letters.

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walaa asakly, Toufik Mansour

Enumeration of compositions according to the sum of the values of the first letters of the occurrences of a 2-letter pattern

Abstract: A composition π = π 1 π 2 • • • π m of a positive integer n is an ordered collection of one or more positive integers whose sum is n. The number of summands, namely m, is called the number of parts of π. We say that π contains a rise, a weak-rise, a level, a descent, or a weak-descent at position i according to whether π i < π i+1 , π i π i+1 , π i = π i+1 , π i > π i+1 , or π i π i+1. Using linear algebra, we determine formulas for generating functions that count compositions of n with m parts, according to the numbers of rises, weak-rises, levels, descents, and weak-descents, and according to the sum, over all occurrences of the rises, weak-rises, levels, descents, and weak-descents, of the first integers in their respective occurrences.

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Maria Verri

A unified approach to the study of general and palindromic compositions

Abstract: We study many properties of compositions of integers by using the symbolic method, multivariate generating functions and Riordan arrays. In particular, we study palindromic compositions with respect to various parameters and present a bijection between walks, compositions and bit strings. The results obtained for compositions can thus be exported to the corresponding objects. We also present a conjecture about a combinatorial interpretation of an algebraic relation involving the number of compositions and the square of the number of palindromic compositions.

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Mark Shattuck, Toufik Mansour

Chebyshev polynomials and statistics on a new collection of words in the Catalan family

Abstract: Recently, a new class of words, denoted by Ln, was shown to be in bijection with a subset of the Dyck paths of length 2n having cardinality the Catalan number C n−1 . Here, we consider statistics on Ln recording the number of occurrences of a letter i. In the cases i = 0 and i = 1, we are able to determine explicit expressions for the number of members of Ln containing a given number of zeros or ones, which generalizes the prior result. To do so, we make use of recurrences to derive a functional equation satisfied by the generating function, which we solve by a new method employing Chebyshev polynomials. In the case i = 0, our result is equivalent to a prior one concerning the distribution of the initial rise statistic on Dyck paths. Recurrences and generating function formulas are also provided in the case of general i.

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James Madden

A Generating Function for the Distribution of Runs in Binary Words

Abstract: Let $N(n,r,k)$ denote the number of binary words of length $n$ that begin with $0$ and contain exactly $k$ runs (i.e., maximal subwords of identical consecutive symbols) of length $r$. We show that the generating function for the sequence $N(n,r,0)$, $n=0,1,\ldots$, is $(1-x)(1-2x + x^r-x^{r+1})^{-1}$ and that the generating function for $\{N(n,r,k)\}$ is $x^{kr}$ time the $k+1$ power of this. We extend to counts of words containing exactly $k$ runs of $1$s by using symmetries on the set of binary words.

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