Abstract

Berget and Rhoades asked whether the permutation representation obtained by the action of $S_{n-1}$ on parking functions of length $n-1$ can be extended to a permutation action of $S_{n}$. We answer this question in the affirmative. We realize our module in two different ways. The first description involves binary Lyndon words and the second involves the action of the symmetric group on the lattice points of the trimmed standard permutahedron.

Highlights

  • Recall that a parking function of length n − 1 is a sequence of nonnegative integers (a1, . . . , an−1) whose nondecreasing rearrangement (b1, . . . , bn−1) satisfies bi i − 1

  • It is immediate that the set of parking functions of length n − 1 carries a permutation action of Sn−1

  • In their study of an extension of Parkn−1, Berget and Rhoades asked [5, Section 4] whether the permutation action of Sn−1 on parking functions of length n − 1 could be extended to a permutation action of Sn

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Summary

Introduction

Recall that a parking function of length n − 1 is a sequence of nonnegative integers (a1, . . . , an−1) whose nondecreasing rearrangement (b1, . . . , bn−1) satisfies bi i − 1. In their study of an extension of Parkn−1, Berget and Rhoades asked [5, Section 4] whether the permutation action of Sn−1 on parking functions of length n − 1 could be extended to a permutation action of Sn. It is worth remarking that the study of “hidden” actions of a larger symmetric group on spaces where a smaller symmetric group acts naturally has received a fair amount of attention in recent years; see for instance [7, 8, 9, 13, 20]. If σ is the identity permutation, Corollary 1.2 says that normalized volume of the standard permutahedron in Rn is equal to the number of lattice points in Pδn. The former is well known to equal nn−2 [18].

The setup
The trimmed standard permutahedron

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