On congruence schemes for constant terms and their applications

TL;DR

This paper revisits and extends the algorithmic framework for analyzing the modulo $p^r$ reductions of combinatorial sequences expressed as constant terms, introducing a new 'scaling' scheme that combines advantages of existing methods.

Contribution

It provides additional details on $p$-schemes, introduces a new combined scheme type, and demonstrates its utility on Motzkin numbers.

Findings

Bounded the number of states in the schemes

Introduced a third scheme type combining automatic and linear benefits

Confirmed and extended a conjecture on Motzkin numbers

Abstract

Rowland and Zeilberger devised an approach to algorithmically determine the modulo $p^r$ reductions of values of combinatorial sequences representable as constant terms (building on work of Rowland and Yassawi). The resulting $p$-schemes are systems of recurrences and, depending on their shape, are classified as automatic or linear. We revisit this approach, provide some additional details such as bounding the number of states, and suggest a third natural type of scheme that combines benefits of automatic and linear ones. We illustrate the utility of these "scaling" schemes by confirming and extending a conjecture of Rowland and Yassawi on Motzkin numbers.