Counting odd numbers in truncations of Pascal's triangle

TL;DR

This paper explores various truncations of Pascal's triangle, analyzing their structure, and characterizes when the number of odd entries per row is a power of two using Lucas's theorem.

Contribution

It introduces a family of Pascal's triangle truncations, generalizes Catalan triangles, and applies Lucas's theorem to identify when odd entries per row are powers of two.

Findings

Identifies specific truncations with odd entries as powers of two.

Provides combinatorial interpretations via lattice paths and tableaux.

Uses Lucas's theorem to characterize these truncations.

Abstract

A "truncation" of Pascal's triangle is a triangular array of numbers that satisfies the usual Pascal recurrence but with a boundary condition that declares some terminal set of numbers along each row of the array to be zero. Presented here is a family of natural truncations of Pascal's triangle that generalize a kind of Catalan triangle. The numbers in each array are realized as differences of binomial coefficients, as counts of certain lattice paths and tableaux, and as entries of representing matrices for certain linear transformations of polynomial spaces. Lucas's theorem is applied to determine precisely those truncations for which the number of odd entries on each row is a power of two.