ALGEBRAIC COMBINATORICS

no. 4
On power ideals of transversal matroids and their “parking functions”
Sarmiento, Camilo1
1 Universidad del Norte Departamento de Matemáticas y Estadística Barranquilla Colombia
Algebraic Combinatorics, Volume 2 (2019) no. 4, pp. 573-583.

To a vector configuration one can associate a polynomial ideal generated by powers of linear forms, known as a power ideal, which exhibits many combinatorial features of the matroid underlying the configuration.

In this note we observe that certain power ideals associated to transversal matroids are, somewhat unexpectedly, monomial. Moreover, the (monomial) basis elements of the quotient ring defined by such a power ideal can be naturally identified with the lattice points of a remarkable convex polytope: a polymatroid, also known as generalized permutohedron. We dub the exponent vectors of these monomial basis elements “parking functions” of the corresponding transversal matroid.

We highlight the connection between our investigation and Stanley–Reisner theory, and relate our findings to Stanley’s conjectured necessary condition on matroid h-vectors.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/alco.57
Classification: 05E40, 05E45, 05B35, 52B40
Keywords: transversal matroids, power ideals, polymatroids, parking functions
Sarmiento, Camilo 1

1 Universidad del Norte Departamento de Matemáticas y Estadística Barranquilla Colombia
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Sarmiento, Camilo. On power ideals of transversal matroids and their “parking functions”. Algebraic Combinatorics, Volume 2 (2019) no. 4, pp. 573-583. doi : 10.5802/alco.57. https://alco.centre-mersenne.org/articles/10.5802/alco.57/

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