Bijection between trees in Stanley character formula and factorizations of a cycle

Abstract

Stanley and Féray gave a formula for the irreducible character of the symmetric group related to a multi-rectangular Young diagram. This formula shows that the character is a polynomial in the multi-rectangular coordinates and gives an explicit combinatorial interpretation for its coefficients in terms of counting certain decorated maps (i.e., graphs drawn on surfaces). In the current paper we concentrate on the coefficients of the top-degree monomials in the Stanley character polynomial, which corresponds to counting certain decorated plane trees. We give an explicit bijection between such trees and minimal factorizations of a cycle.

Figures (28)

• Figure 1: Multi-rectangular Young diagram $\mathbf{p}\times \mathbf{q}=(2,3,1) \times (5,4,2)$.
• Figure 2: An example of a Stanley tree of type $(3,5,3)$. The circled numbers indicate the labels of the black vertices. The black numbers indicate the labels of the edges. The colors (blue for $1$, red for $2$, green for $3$) indicate the values of the function $f$ on white vertices.
• Figure 3: One of the plane trees that correspond to the minimal factorization $\sigma_1 \cdots \sigma_n= \sigma_1 \sigma_2 \sigma_3 \sigma_4 = (1,2,\dots,10)\in\mathfrak{S}_{k}$ of a long cycle with $\sigma_1=(7,8,9,10)$, $\sigma_2=(1,2)$, $\sigma_3=(2,5,6,10)$, $\sigma_4=(2,3,4)$, $k=10$ and $n=4$. The black vertices correspond to the cycles $\sigma_1,\dots,\sigma_4$. The white vertices correspond to the elements of the set $\{1,\dots,10\}$ on which acts the symmetric group $\mathfrak{S}_{10}$. The cyclic order of the edges around the black vertices is determined by the cycles $\sigma_1,\dots,\sigma_4$. The cyclic order of the edges around the white vertices is arbitrary.
• Figure 4: The structure of a white spine vertex $c$ in the plane tree $T_0$. The labels of the black spine vertices $\alpha_c,y_1\in\{1,\dots,n\}$ fulfill $\alpha_c< y_1$. There are $l\geq 0$ non-spine neighbors of the vertex $c$, denoted $x_1,\dots,x_l$. They are all placed (in the counterclockwise cyclic order) after $\alpha_c$ and before $y_1$. It may happen that $l=0$ and there are no non-spine neighbors of $c$. For the details see \ref{['sec:tt0']}.
• Figure 5: \ref{['subfig:ex1']} The output $T_0$ of the first step of the algorithm $\mathcal{A}$ applied to the minimal factorization \ref{['eq:example']}. The spine is the horizontal red path between the vertices $1$ and $14$. \ref{['subfig:ex2']} The tree $T_1$ with some sample clusters highlighted. The red color indicates the cluster $1$, while the green indicates the cluster $15$. The double transverse lines identify the roots of the respective clusters.

Theorems & Definitions (25)

• Example 1
• Lemma 1.1
• Example 2
• Corollary 1.2
• Remark 1.3
• Theorem 2.1
• Lemma 2.3
• Lemma 2.4
• Example 3
• Proposition 2.5