https://github.com/debdut/rational-parking-functions
https://github.com/debdut/rational-parking-functions
Last synced: about 2 months ago
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- Host: GitHub
- URL: https://github.com/debdut/rational-parking-functions
- Owner: Debdut
- Created: 2020-01-24T20:14:34.000Z (over 6 years ago)
- Default Branch: master
- Last Pushed: 2020-01-24T23:49:18.000Z (over 6 years ago)
- Last Synced: 2025-03-03T02:41:46.903Z (about 1 year ago)
- Language: Python
- Size: 6.84 KB
- Stars: 3
- Watchers: 2
- Forks: 0
- Open Issues: 0
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Metadata Files:
- Readme: README.md
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README
This code is build-up and experiments for my paper published at Electronic Journal Of Combinatorics
https://www.combinatorics.org/ojs/index.php/eljc/article/download/v25i1p21/pdf/
***Abstract***
The classical parking functions, counted by the Cayley number (n+1)^(n-1), carry a natural permutation representation of the symmetric group S_n in which the number of orbits is the n'th Catalan number. In this paper, we will generalize this setup to rational parking functions indexed by a pair (a,b) of coprime positive integers. We show that these parking functions, which are counted by b^(a-1), carry a permutation representation of S_a in which the number of orbits is a rational Catalan number. We compute the Frobenius characteristic of the S_a-module of (a,b)-parking functions. Next we propose a combinatorial formula for a q-analogue of the rational Catalan numbers and relate this formula to a new combinatorial model for q-binomial coefficients. Finally, we discuss q,t-analogues of rational Catalan numbers and parking functions (generalizing the shuffle conjecture for the classical case) and present several conjectures.