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A conjectured formula for the rational $q,t$-Catalan polynomial | | Graham Hawkes
; | | Date: |
1 Aug 2022 | | Abstract: | We conjecture a formula for the rational $q,t$-Catalan polynomial
$mathcal{C}_{r/s}$ that is symmetric in $q$ and $t$ by definition. Denoting by
$mathcal{C}_{r/s}^d$ the homogeneous part of degree $d$ less than the maximum
degree appearing in $mathcal{C}_{r/s}$, we prove that our conjecture is
correct on the set ${mathcal{C}_{r/s}^d: r cong 1mod s, , d leq 20}$. In
the process we show that for any finite $d^*$ providing a combinatorial proof
of the symmetry of the infinite set of functions ${mathcal{C}_{r/s}^d: r
cong 1mod s, , d leq d^*}$ is equivalent to carrying out a finite number
of base case computations depending only on $d^*$. We provide python code
needed to carry out these computations for $d^*=20$ (or any finite $d^*$) as
well as python code that can be used to check the conjecture for any relatively
prime $(s,r)$ for all $d$. | | Source: | arXiv, 2208.00577 | | Services: | Forum | Review | PDF | Favorites | |
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