Rational Catalan polynomials and rank words
| Autor: | Kaliszewski, Ryan, Li, Huilan |
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| Rok vydání: | 2016 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | For $m,n$ coprime we introduce a new statistic skip on $(m,n)$-rational Dyck paths and give a fast way to compute dinv and skip statistics. We also introduce $(m,n)$-rank words, which are in one-to-one correspondence with $(m,n)$-Dyck paths. Defining an equivalence relation on pairs of certain ranks in a rank word, we prove that the number of equivalence classes is the skips of the rank word, and the skips of the corresponding Dyck path. We construct a homogeneous generating function $W_{m,n}(q,t,b)$ using statistics area, dinv and skip, where $W_{m,n}(q,t,1)=C_{m,n}(q,t)$, the rational Catalan polynomial. We then give an explicit formula for $(3,n)$-rational Catalan polynomials and prove they are $q,t$-symmetric. Comment: 18 pages; continuation of work presented in FPSAC 2015 |
| Databáze: | arXiv |
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