Rational Parking Functions and Catalan Numbers
Rational Parking Functions and Catalan Numbers
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合理的停车功能和加泰罗尼亚数字
DOI：
10.1007/s00026-015-0293-6
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发表时间：
2014
影响因子：
0.5
通讯作者：
G. Warrington
中科院分区：
文献类型：
--
作者：
D. Armstrong;N. Loehr;G. Warrington
“经典”的停车函数，由Cayley数（n+1）n−1计算，携带对称群Sn的自然排列表示，其中轨道数为加泰罗尼亚数$${\frac{1}{n+1} \left( \begin{array}{ll} 2n \\ n \end{array} \right)}$$ 1n+1（2nn）。在本文中，我们将这种建立推广到由一对（a, b）的素数正整数索引的“有理”停放函数。这些用ba−1计数的停车函数携带了一个Sa的排列表示，其中轨道的数量是“有理数”加泰罗尼亚数$${\frac{1}{a+b} \left( \begin{array}{ll} a+b \\ a \end{array} \right)}$$ 1a+ba+ba。首先，我们计算了(a, b)-停车函数的sa模的Frobenius特征，给出了这个对称函数在完全齐次基、幂和基和Schur基下的显式展开式。其次，我们研究了有理加泰罗尼亚数的q-类似物，推测了有理q-加泰罗尼亚数$${\frac{1}{[a+b]_{q}} {{\left[ \begin{array}{ll} a+b \\ a \end{array} \right]}_{q}}}$$ 1[a+b]qa+baq和q-二项式系数$${{{\left[ \begin{array}{ll} n \\ k \end{array} \right]}_{q}}}$$ nkq的新的组合公式。我们给出了用[a+b]除q的双射解释，证明了这两个猜想的等价性。第三，我们给出了有理加泰罗尼亚数的q、t类似物和停放函数的组合定义，推广了经典情况下的Shuffle猜想。我们提出了关于这些多项式的联合对称和t = 1/q专门化的几个猜想。附录对a和b的小值明确地计算了这些多项式。
The “classical” parking functions, counted by the Cayley number (n+1)n−1, carry a natural permutation representation of the symmetric group Sn in which the number of orbits is the Catalan number $${\frac{1}{n+1} \left( \begin{array}{ll} 2n \\ n \end{array} \right)}$$1n+1(2nn). In this paper, we will generalize this setup to “rational” parking functions indexed by a pair (a, b) of coprime positive integers. These parking functions, which are counted by ba−1, carry a permutation representation of Sa in which the number of orbits is the “rational” Catalan number $${\frac{1}{a+b} \left( \begin{array}{ll} a+b \\ a \end{array} \right)}$$1a+ba+ba. First, we compute the Frobenius characteristic of the Sa-module of (a, b)-parking functions, giving explicit expansions of this symmetric function in the complete homogeneous basis, the power-sum basis, and the Schur basis. Second, we study q-analogues of the rational Catalan numbers, conjecturing new combinatorial formulas for the rational q-Catalan numbers $${\frac{1}{[a+b]_{q}} {{\left[ \begin{array}{ll} a+b \\ a \end{array} \right]}_{q}}}$$1[a+b]qa+baq and for the q-binomial coefficients $${{{\left[ \begin{array}{ll} n \\ k \end{array} \right]}_{q}}}$$nkq. We give a bijective explanation of the division by [a+b]q that proves the equivalence of these two conjectures. Third, we present combinatorial definitions for q, t-analogues of rational Catalan numbers and parking functions, generalizing the Shuffle Conjecture for the classical case. We present several conjectures regarding the joint symmetry and t = 1/q specializations of these polynomials. An appendix computes these polynomials explicitly for small values of a and b.