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.

Revised:

Accepted:

Published online:

Keywords: $h$-positivity, Lyndon word, parking function, permutahedron.

^{1}; Sulzgruber, Robin

^{2}; Tewari, Vasu

^{3}

@article{ALCO_2021__4_4_663_0, author = {Konvalinka, Matja\v{z} and Sulzgruber, Robin and Tewari, Vasu}, title = {Trimming the permutahedron to extend the parking space}, journal = {Algebraic Combinatorics}, pages = {663--674}, publisher = {MathOA foundation}, volume = {4}, number = {4}, year = {2021}, doi = {10.5802/alco.173}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/alco.173/} }

TY - JOUR AU - Konvalinka, Matjaž AU - Sulzgruber, Robin AU - Tewari, Vasu TI - Trimming the permutahedron to extend the parking space JO - Algebraic Combinatorics PY - 2021 SP - 663 EP - 674 VL - 4 IS - 4 PB - MathOA foundation UR - http://archive.numdam.org/articles/10.5802/alco.173/ DO - 10.5802/alco.173 LA - en ID - ALCO_2021__4_4_663_0 ER -

%0 Journal Article %A Konvalinka, Matjaž %A Sulzgruber, Robin %A Tewari, Vasu %T Trimming the permutahedron to extend the parking space %J Algebraic Combinatorics %D 2021 %P 663-674 %V 4 %N 4 %I MathOA foundation %U http://archive.numdam.org/articles/10.5802/alco.173/ %R 10.5802/alco.173 %G en %F ALCO_2021__4_4_663_0

Konvalinka, Matjaž; Sulzgruber, Robin; Tewari, Vasu. Trimming the permutahedron to extend the parking space. Algebraic Combinatorics, Volume 4 (2021) no. 4, pp. 663-674. doi : 10.5802/alco.173. http://archive.numdam.org/articles/10.5802/alco.173/

[1] Canonical representatives for divisor classes on tropical curves and the matrix-tree theorem, Forum Math. Sigma, Volume 2 (2014), e24, 25 pages | DOI | MR | Zbl

[2] The equivariant volumes of the permutahedron, Sém. Lothar. Combin., Volume 82B (2020), 16, 12 pages | MR | Zbl

[3] Riemann–Roch theory for graph orientations, Adv. Math., Volume 309 (2017), pp. 655-691 | DOI | MR | Zbl

[4] Riemann–Roch and Abel–Jacobi theory on a finite graph, Adv. Math., Volume 215 (2007) no. 2, pp. 766-788 | DOI | MR | Zbl

[5] Extending the parking space, J. Combin. Theory Ser. A, Volume 123 (2014), pp. 43-56 | DOI | MR | Zbl

[6] An extension of a formula of Jovovic, Integers, Volume 19 (2019), A47, 7 pages | MR | Zbl

[7] Symmetric group actions on the cohomology of configurations in ${\mathbb{R}}^{d}$, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl., Volume 21 (2010) no. 3, pp. 235-250 | DOI | MR | Zbl

[8] On configuration spaces and Whitehouse’s lifts of the Eulerian representations, J. Pure Appl. Algebra, Volume 223 (2019) no. 10, pp. 4524-4535 | DOI | MR | Zbl

[9] The actions of ${S}_{n+1}$ and ${S}_{n}$ on the cohomology ring of a Coxeter arrangement of type ${A}_{n-1}$, Manuscripta Math., Volume 91 (1996) no. 1, pp. 83-94 | DOI | MR | Zbl

[10] Conjectures on the quotient ring by diagonal invariants, J. Algebraic Combin., Volume 3 (1994) no. 1, pp. 17-76 | DOI | MR | Zbl

[11] Some natural extensions of the parking space, J. Combin. Theory Ser. A, Volume 180 (2021), 105394, 19 pages | DOI | MR | Zbl

[12] Combinatorics on words, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1997, xviii+238 pages (With a foreword by Roger Lyndon and a preface by Dominique Perrin, Corrected reprint of the 1983 original, with a new preface by Perrin) | DOI | MR | Zbl

[13] Hidden ${\Sigma}_{n+1}$-actions, Comm. Math. Phys., Volume 176 (1996) no. 2, pp. 467-474 | MR | Zbl

[14] Permutohedra, associahedra, and beyond, Int. Math. Res. Not. IMRN (2009) no. 6, pp. 1026-1106 | DOI | MR | Zbl

[15] Trees, parking functions, syzygies, and deformations of monomial ideals, Trans. Amer. Math. Soc., Volume 356 (2004) no. 8, pp. 3109-3142 | DOI | MR | Zbl

[16] An inequality, J. London Math. Soc., Volume 27 (1952), pp. 1-6 | DOI | MR

[17] Aspects of the topology and combinatorics of Higgs bundle moduli spaces, SIGMA Symmetry Integrability Geom. Methods Appl., Volume 14 (2018), 18 pages | DOI | MR | Zbl

[18] A zonotope associated with graphical degree sequences, Applied geometry and discrete mathematics (DIMACS Ser. Discrete Math. Theoret. Comput. Sci.), Volume 4, Amer. Math. Soc., Providence, RI, 1991, pp. 555-570 | MR | Zbl

[19] Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, 62, Cambridge University Press, Cambridge, 1999, xii+581 pages (With a foreword by Gian-Carlo Rota and appendix 1 by Sergey Fomin) | DOI | MR | Zbl

[20] The Eulerian representations of ${\Sigma}_{n}$ as restrictions of representations of ${\Sigma}_{n+1}$, J. Pure Appl. Algebra, Volume 115 (1997) no. 3, pp. 309-320 | DOI | MR | Zbl

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