On a New Congruence in the Catalan Triangle
Jovan Mikić
TL;DR
This work introduces an alternating convolution $S(2n,m)$ in the Catalan triangle and proves its divisibility by both the Fuss-Catalan number $C^{(3)}_n$ and the central binomial coefficient $\binom{2n}{n}$. Central to the approach is the novel framework of $M$ sums, together with the integer quantity $T(n,j)$, which the authors interpret combinatorially in terms of lattice paths and connect to Schröder numbers of order two. The paper provides explicit formulas for key sums, notably $S(2n,1)=(-1)^n\,C^{(3)}_n\,C_n\,(2n^2+n+1)$, and develops a suite of auxiliary divisibility lemmas to establish the general $m$-dependent results. Beyond the core Catalan-triangle results, the authors derive closed forms for Schröder-type path counts, leading to a JS-type summation formula, thus linking Catalan, Fuss-Catalan, and Schröder combinatorics in a unified framework with potential implications for lattice-path enumeration and modular properties of combinatorial sums.
Abstract
For $0\leq k \leq n$, the number $C(n,k)$ represents the number of all lattice paths in the plane from the point $(0,0)$ to the point $(n,k)$, using steps $(1,0)$ and $(0,1)$, that never rise above the main diagonal $y=x$. The Fuss-Catalan number of order three $C^{(3)}_n$ represents the number of all lattice paths in the plane from the point $(0,0)$ to the point $(2n,n)$, using steps $(1,0)$ and $(0,1)$, that do not rise above the line $y=\frac{x}{2}$. We present a new alternating convolution formula for the numbers $C(2n,k)$. By using a new class of binomial sums that we call $M$ sums, we prove that this sum is divisible by $C^{(3)}_n$ and by the central binomial coefficient $\binom{2n}{n}$. We do this by examining the numbers $T(n,j)=\frac{1}{2n+1}\binom{2n+j}{j}\binom{2n+1}{n+j+1}$, for which we present a new combinatorial interpretation, connecting them to the generalized Schröder numbers of order two.