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Directed lattice paths avoiding periodic subset of points on "time"-axis

S. Tarasov

TL;DR

The paper addresses the enumeration of directed lattice paths that avoid a periodic subset of the time-axis by developing a generating-function framework. It reduces the counting problem to a finite linear system with a circulant coefficient matrix when the forbidden set is periodic, and relates system coefficients to loop counts via $^dL_k$ and $^dSL_k$, enabling explicit expressions through loop Generating Functions. A key result is the proof of the Hajnal–Nagy conjecture, established through a determinant identity derived from the circulant structure, linking restricted path counts to classical loop statistics and Polya-type return probabilities. The approach provides a unified, algebraic method to compute restricted path generating functions in arbitrary dimension and reveals deep connections between path restrictions, circulant matrices, and lattice-loop theory, with potential implications for combinatorial identities and exact counting in related models.

Abstract

We compute generating functions of the set of directed lattice paths starting from the origin and avoiding a periodic set of even point on OX = "time"-axis. As an application we prove a combinatorial identity proposed by P. Hajnal and G.V. Nagy.

Directed lattice paths avoiding periodic subset of points on "time"-axis

TL;DR

The paper addresses the enumeration of directed lattice paths that avoid a periodic subset of the time-axis by developing a generating-function framework. It reduces the counting problem to a finite linear system with a circulant coefficient matrix when the forbidden set is periodic, and relates system coefficients to loop counts via and , enabling explicit expressions through loop Generating Functions. A key result is the proof of the Hajnal–Nagy conjecture, established through a determinant identity derived from the circulant structure, linking restricted path counts to classical loop statistics and Polya-type return probabilities. The approach provides a unified, algebraic method to compute restricted path generating functions in arbitrary dimension and reveals deep connections between path restrictions, circulant matrices, and lattice-loop theory, with potential implications for combinatorial identities and exact counting in related models.

Abstract

We compute generating functions of the set of directed lattice paths starting from the origin and avoiding a periodic set of even point on OX = "time"-axis. As an application we prove a combinatorial identity proposed by P. Hajnal and G.V. Nagy.

Paper Structure

This paper contains 6 sections, 3 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. The main theorem
  3. Loops. Circulants. Proof of HN-conjecture
  4. Loops
  5. Circulants
  6. Proof HN-conjecture

Key Result

Theorem 1

Let $\mathbf{A} = (A=\{a_0(=0),\dots,a_k\}, t_A)$ be periodic. The set of generating functions $^dP^r(\mathbf{A} ,t), r\in A$ of DLP's on $\mathbb Z_+\times\mathbb Z^d$ satisfies the following system of linear equations in the ring of formal power series $\mathbb Q[\![t]\!]$: Moreover system has a unique solution.

Theorems & Definitions (6)

  • Remark 1
  • Theorem 1
  • proof
  • Proposition 1
  • Lemma 1
  • proof