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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2503.05013 |
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| _version_ | 1866912263629176832 |
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| author | Mikić, Jovan |
| author_facet | Mikić, Jovan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_05013 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On a New Congruence in the Catalan Triangle Mikić, Jovan Combinatorics 05A10, 05A19 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. |
| title | On a New Congruence in the Catalan Triangle |
| topic | Combinatorics 05A10, 05A19 |
| url | https://arxiv.org/abs/2503.05013 |