Automated Discovery and Proof of Congruence Theorems for Partial Sums of Combinatorial Sequences

TL;DR

This paper presents an automated method to discover and prove congruence theorems for partial sums of combinatorial sequences expressed as constant terms of Laurent polynomial powers, applicable to any prime and sum length.

Contribution

It introduces a general formula for congruences of partial sums of combinatorial sequences and automates their discovery and proof using computer algebra.

Findings

Automated discovery of congruence theorems for combinatorial sums.

The set of residues modulo p is often finite, independent of p.

The method applies to sequences expressed as constant terms of Laurent polynomial powers.

Abstract

Many combinatorial sequences (for example, the Catalan and Motzkin numbers) may be expressed as the constant term of $P(x)^k Q(x)$, for some Laurent polynomials $P(x)$ and $Q(x)$ in the variable $x$ with integer coefficients. Denoting such a sequence by $a_k$, we obtain a general formula that determines the congruence class, modulo $p$, of the indefinite sum $\sum_{k=0}^{rp -1} a_k$, for {\it any} prime $p$, and any positive integer $r$, as a linear combination of sequences that satisfy linear recurrence (alias difference) equations with constant coefficients. This enables us (or rather, our computers) to automatically discover and prove congruence theorems for such partial sums. Moreover, we show that in many cases, the set of the residues is finite, regardless of the prime $p$.