A simpler formula for the number of diagonal inversions of an (m, n)-parking function and a returning fermionic formula.
Discrete Mathematics(2015)
摘要
Recent results have placed the classical shuffle conjecture of Haglund et al. in a broader context of an infinite family of conjectures about parking functions in any rectangular lattice. The combinatorial side of the new conjectures has been defined using a complicated generalization of the dinv statistic that is composed of three parts and that is not obviously non-negative. Here we simplify the definition of dinv, prove that it is always non-negative, and give a graphical description of the statistic in the style of the classical case. We go on to show that in the (n−1)×n lattice, parking functions satisfy a fermionic formula that is similar to the one given in the classical case by Haglund and Loehr.
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关键词
Rational parking functions,Shuffle conjecture,Dinv,Diagonal inversions,Fermionic formula
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