论文标题
在签名的多重性上，围绕培养皿对称函数的Schur扩展
On Signed Multiplicities of Schur Expansions Surrounding Petrie Symmetric Functions
论文作者
论文摘要
对于$ k \ ge 1 $，同质对称函数$ g（k，m）$ $ m $ $ m $由$ \ sum_ {m \ ge 0} g（k，m）z^m = \ prod_ {i \ ge 1}称为\ emph {petrie对称函数}。正如Grinberg和Fu--Mei独立得出的那样，在Schur函数的基础上，$ G（K，M）$的扩展$S_λ$变成了免费签名的多重性，即系数为$ -1 $，$ 0 $ $ 0 $和$ 1 $。在本文中，我们就$λ$的$ k $ core和一系列从$λ$中取出的$λ$的$ k $ core的系数进行了组合解释。我们进一步研究了$ g（k，m）$的产品，并具有功率和对称函数$ p_n $。对于所有$ n \ ge 1 $，我们在$ g（k，m）\ cdot p_n $扩展的顺序上提供了必要的条件，以签署多重性签名的$ g（k，m）\ cdot p_n $。这可以肯定地解决亚历山大森的猜想，因为特殊情况$ n = 2 $。
For $k\ge 1$, the homogeneous symmetric functions $G(k,m)$ of degree $m$ defined by $\sum_{m\ge 0} G(k,m) z^m=\prod_{i\ge 1} \big(1+x_iz+x^2_iz^2+\cdots+x^{k-1}_iz^{k-1}\big)$ are called \emph{Petrie symmetric functions}. As derived by Grinberg and Fu--Mei independently, the expansion of $G(k,m)$ in the basis of Schur functions $s_λ$ turns out to be signed multiplicity free, i.e., the coefficients are $-1$, $0$ and $1$. In this paper we give a combinatorial interpretation of the coefficient of $s_λ$ in terms of the $k$-core of $λ$ and a sequence of rim hooks of size $k$ removed from $λ$. We further study the product of $G(k,m)$ with a power sum symmetric function $p_n$. For all $n\ge 1$, we give necessary and sufficient conditions on the parameters $k$ and $m$ in order for the expansion of $G(k,m)\cdot p_n$ in the basis of Schur functions to be signed multiplicity free. This settles affirmatively a conjecture of Alexandersson as the special case $n=2$.
