插座连接系数的recurrence formula
A recurrence formula for Jack connection coefficients
摘要：
本文研究了与Jack对称函数理论密切相关的对称群的群代数的经典交换子代数的联系系数--Jack联系系数.首先由Goulden和Jackson（1996）引入的这些数由给定整数$n$和Jack参数$α$的三个分区索引，定义为Jack对称函数的柯西和的幂和展开中的系数。虽然我们对它们知之甚少，但$n$的小值计算的例子表明，特殊情况$α=1$（类代数的连接系数）和$α= 2$（双陪集代数的连接系数）的良好性质扩展到一般的$α$。Goulden和Jackson证明了Jack连接系数是$β= α-1$中的多项式，其非负整数系数由一组$2n$元素上的匹配的统计量给出，即所谓的Matchings-Jack猜想.在本文中，我们研究了两个整数分割等于单个部分$(n)$的情况，并使用Lasalle（2008）的Jack对称函数框架来证明系数满足一个简单的递归公式，这使得它们的计算非常有效，并允许更好地理解它们的性质。特别是，我们证明了匹配杰克猜想在这种情况下。此外，我们提供了一个双射证明的递归公式$α\in \{1,2\}$使用组合解释的系数，这些特定值的杰克参数。最后，我们展示了多项式性质的更一般的系数，其中两个单一的部分分区被替换为任意数量的整数分区等于$(n)$或$[1^{n-2}2]$。
英文摘要：
This article is devoted to the study of Jack connection coefficients, a generalization of the connection coefficients of the classical commutative subalgebras of the group algebra of the symmetric group closely related to the theory of Jack symmetric functions. First introduced by Goulden and Jackson (1996) these numbers indexed by three partitions of a given integer $n$ and the Jack parameter $α$ are defined as the coefficients in the power sum expansion of the Cauchy sum for Jack symmetric functions. While very little is known about them, examples of computations for small values of $n$ tend to show that the nice properties of the special cases $α=1$ (connection coefficients of the class algebra) and $α= 2$ (connection coefficients of the double coset algebra) extend to general $α$. Goulden and Jackson conjectured that Jack connection coefficients are polynomials in $β= α-1$ with non negative integer coefficients given by some statistics on matchings on a set of $2n$ elements, the so called Matchings-Jack conjecture. In this paper we look at the case when two of the integer partitions are equal to the single part $(n)$ and use a framework by Lasalle (2008) for Jack symmetric functions to show that the coefficients satisfy a simple recurrence formula that makes their computation very effective and allow a better understanding of their properties. In particular we prove the Matchings-Jack conjecture in this case. Furthermore, we provide a bijective proof of the recurrence formula for $α\in \{1,2\}$ using the combinatorial interpretation of the coefficients for these specific values of the Jack parameter. Finally we exhibit the polynomial properties of more general coefficients where the two single part partitions are replaced by an arbitrary number of integer partitions either equal to $(n)$ or $[1^{n-2}2]$.
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