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Generating functions for compositions with constrained even parts

Mahdi Koutchoukali

Abstract

We study compositions of a positive integer $n$ in which the occurrence of even parts larger than a fixed threshold $k$ is controlled. More precisely, for each composition $m=(m_1,\dots,m_r)$ we consider the number of even parts strictly larger than $k$, and we introduce a two-variable generating function that encodes this statistic. We show that this generating function is rational and obtain explicit closed forms, depending on the parity of $k$. As a consequence, we derive exact counting formulas and linear recurrence relations for the number of compositions of $n$ with a prescribed number of even parts greater than $k$. We also obtain explicit formulas for related refined quantities, such as the number of compositions with an even or odd number of such parts, the total number of their occurrences among all compositions of $n$, and positional statistics describing how late the first such part appears in a composition. This combinatorial problem is motivated by questions arising from combinatorial expansions related to zeta functions of algebraic curves over finite fields, although the results of this paper are entirely combinatorial.

Generating functions for compositions with constrained even parts

Abstract

We study compositions of a positive integer in which the occurrence of even parts larger than a fixed threshold is controlled. More precisely, for each composition we consider the number of even parts strictly larger than , and we introduce a two-variable generating function that encodes this statistic. We show that this generating function is rational and obtain explicit closed forms, depending on the parity of . As a consequence, we derive exact counting formulas and linear recurrence relations for the number of compositions of with a prescribed number of even parts greater than . We also obtain explicit formulas for related refined quantities, such as the number of compositions with an even or odd number of such parts, the total number of their occurrences among all compositions of , and positional statistics describing how late the first such part appears in a composition. This combinatorial problem is motivated by questions arising from combinatorial expansions related to zeta functions of algebraic curves over finite fields, although the results of this paper are entirely combinatorial.
Paper Structure (16 sections, 15 theorems, 116 equations)

This paper contains 16 sections, 15 theorems, 116 equations.

Table of Contents

  1. Introduction
  2. Notations and basic definitions
  3. The main generating function
  4. An explicit rational form
  5. A recurrence for the coefficients
  6. Specializations and refined enumerations
  7. Avoiding large even parts: the specialization $y=0$
  8. Even versus odd number of large even parts
  9. Total number of large even parts among all compositions
  10. Positional statistics: Late appearance of large even parts
  11. Generating functions for allowed and forbidden parts.
  12. Compositions whose first $\ell$ parts avoid $\mathrm{Ev}_{>k}$.
  13. First large even part at position exactly $\ell+1$.
  14. Removing the "no large even part at all" case.
  15. Explicit formula for $L_{k,\ell}(n)$
  16. ...and 1 more sections

Key Result

Proposition 1

Let $k\ge 1$ be fixed. Then the generating function is a rational function in $x$ and $y$. More precisely, one has where the weights $w_j$ are given by Equivalently,

Theorems & Definitions (35)

  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Corollary 1
  • Proposition 3
  • proof
  • Example 1
  • Proposition 4
  • proof
  • ...and 25 more