The Catalan matroid

Ardila, Federico

doi:10.1016/s0097-3165(03)00121-3

Cited by 37 publications

(60 citation statements)

References 9 publications

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“…More generally, we may consider the lattice paths of 2n steps (1, 1) and (1, −1) which stay between an upper and a lower border; their upstep sets form the bases of a lattice path matroid. [Ard03,BdMN03] 7. (Schubert matroids) Given n ∈ N and a set of positive integers which realizes M as a linear matroid, a graph which realizes it as a graphical matroid, a bipartite graph which realizes it as a transversal matroid, the five Dyck paths of length 3 which show that M is the Catalan matroid C 3 , and a directed graph with sinks B 0 = {a, c, e} which realize M as a cotransversal matroid.…”

Section: Examplesmentioning

confidence: 99%

Algebraic and Geometric Methods in Enumerative Combinatorics

Ardila

2015

Handbook of Enumerative Combinatorics

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Section: Examplesmentioning

confidence: 99%

Algebraic and Geometric Methods in Enumerative Combinatorics

Ardila

2015

Handbook of Enumerative Combinatorics

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“…, 2n}. These matroids are called Catalan matroids and have been study extensively recently, see [5] or [1].…”

Section: Catalan Matroidsmentioning

confidence: 99%

“…And then T M +e (0, 2) = 2T M (1,2), that is twice the number of spanning sets of M . But the number of bases of M + e is just the number of independent sets of size r and r − 1.…”

Section: Catalan Matroidsmentioning

confidence: 99%

A note on some inequalities for the Tutte polynomial of a matroid

Merino

Ibanez

Rodríguez

2009

Electronic Notes in Discrete Mathematics

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We prove that if a matroid M contains two disjoint bases (or, dually, if its ground set is the union of two bases), thenThis resembles the conjecture that appears in C. Merino and D.J.A. Welsh, Forests, colourings and acyclic orientations of the square lattice, Annals of Combinatorics 3 (1999) pp. 417-429: If G is a 2-connected graph with no loops, then T G (1, 1) ≤ max{T G (2, 0), T (0, 2)}. We conjecture that T M (1, 1) ≤ max{T M (2, 0), T M (0, 2)} for matroids which contains two disjoint bases or its ground set is the union of two bases. We also prove the latter for some families of graphs and matroids.

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“…Example 4.12. Because adjoining an isthmus and taking a single-point free extension of a matroid correspond to free multiplication on the right by I and Z, respectively, it follows that the class of matroids introduced in [3], now variously known as generalized Catalan matroids [2], shifted matroids [1] and freedom matroids [5], is the class generated by the single-element matroids under free product.…”

Section: Basic Properties Of the Free Productmentioning

confidence: 99%

A unique factorization theorem for matroids

Crapo

Schmitt

2005

Journal of Combinatorial Theory, Series A

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We study the combinatorial, algebraic and geometric properties of the free product operation on matroids. After giving cryptomorphic definitions of free product in terms of independent sets, bases, circuits, closure, flats and rank function, we show that free product, which is a noncommutative operation, is associative and respects matroid duality. The free product of matroids M and N is maximal with respect to the weak order among matroids having M as a submatroid, with complementary contraction equal to N. Any minor of the free product of M and N is a free product of a repeated truncation of the corresponding minor of M with a repeated Higgs lift of the corresponding minor of N . We characterize, in terms of their cyclic flats, matroids that are irreducible with respect to free product, and prove that the factorization of a matroid into a free product of irreducibles is unique up to isomorphism. We use these results to determine, for K a field of characteristic zero, the structure of the minor coalgebra K{M} of a family of matroids M that is closed under formation of minors and free products: namely, K{M} is cofree, cogenerated by the set of irreducible matroids belonging to M.

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The Catalan matroid

Cited by 37 publications

References 9 publications

Algebraic and Geometric Methods in Enumerative Combinatorics

Algebraic and Geometric Methods in Enumerative Combinatorics

A note on some inequalities for the Tutte polynomial of a matroid

A unique factorization theorem for matroids

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