The Tutte q-Polynomial
Guus Bollen, Henry Crapo, Relinde Jurrius

TL;DR
This paper explores the extension of Tutte polynomials to q-matroids, focusing on their properties, definitions of activity, and the challenges in generalizing classical matroid concepts to the q-analogue.
Contribution
It introduces a framework for defining Tutte polynomials and activities in q-matroids, extending classical matroid theory to a lattice-theoretic context.
Findings
Established a q-analogue of Tutte polynomial computation methods.
Identified the unique clopen flat in prime-free minors of q-matroids.
Outlined the challenges in extending the x→(x-1), y→(y-1) substitution to q-matroids.
Abstract
-Matroids are defined on complemented modular support lattices. Minors of length 2 are of four types as in a "classical" matroid. Tutte polynomials of matroids are calculated either by recursion over deletion/contraction of single elements, by an enumeration of bases with respect to internal/external activities, or by substitution in their rank generating functions . The -analogue of the passage from a Tutte polynomial to its corresponding RGF is straight-forward, but the analogue of the reverse process is more delicate. For matroids on a set , and relative to any linear order on the points, the concept of internal/external activity of a point relative to a basis gives rise to a partition of the underlying Boolean algebra into a set of "prime-free" (or "structureless") minors, such…
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · graph theory and CDMA systems · Advanced Algebra and Logic
