$q$-Super Catalan Numbers: Combinatorial identities, Generating Functions, and Narayana Refinements
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909620749991936 |
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| author | Rodelet--Causse, Arthur Tevlin, Lenny |
| author_facet | Rodelet--Causse, Arthur Tevlin, Lenny |
| contents | We begin by deriving a number of combinatorial identities satisfied by the $q$-super Catalan numbers. In particular, we extend some of the known combinatorial identities (Touchard, Koshy, Reed Dawson) to the $q$-super Catalan numbers. Next, we introduce some $q$-convolution identities involving q-central binomial and q-Catalan numbers and derive a generating function for $q$-Catalan numbers. Then we introduce Narayana-type refinements of the super Catalan numbers. We prove algebraically the $γ$-positivity of those refinements and give a combinatorial proof in a special case through the type B analog of noncrossing partitions. Then we introduce their natural $q$-analogs, prove their $q$-$γ$-positivity and prove some identities they satisfy, generalizing identities of Kreweras and Le Jen-Shoo. Using yet another identity, we prove that these refinements are positive integer polynomials in $q$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_09821 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $q$-Super Catalan Numbers: Combinatorial identities, Generating Functions, and Narayana Refinements Rodelet--Causse, Arthur Tevlin, Lenny Combinatorics Number Theory 05A10, 11B65 We begin by deriving a number of combinatorial identities satisfied by the $q$-super Catalan numbers. In particular, we extend some of the known combinatorial identities (Touchard, Koshy, Reed Dawson) to the $q$-super Catalan numbers. Next, we introduce some $q$-convolution identities involving q-central binomial and q-Catalan numbers and derive a generating function for $q$-Catalan numbers. Then we introduce Narayana-type refinements of the super Catalan numbers. We prove algebraically the $γ$-positivity of those refinements and give a combinatorial proof in a special case through the type B analog of noncrossing partitions. Then we introduce their natural $q$-analogs, prove their $q$-$γ$-positivity and prove some identities they satisfy, generalizing identities of Kreweras and Le Jen-Shoo. Using yet another identity, we prove that these refinements are positive integer polynomials in $q$. |
| title | $q$-Super Catalan Numbers: Combinatorial identities, Generating Functions, and Narayana Refinements |
| topic | Combinatorics Number Theory 05A10, 11B65 |
| url | https://arxiv.org/abs/2505.09821 |