Tiling bijections between paths and Brauer diagrams

Abstract

There is a natural bijection between Dyck paths and basis diagrams of the Temperley-Lieb algebra defined via tiling. Overhang paths are certain generalisations of Dyck paths allowing more general steps but restricted to a rectangle in the two-dimensional integer lattice. We show that there is a natural bijection, extending the above tiling construction, between overhang paths and basis diagrams of the Brauer algebra.Comment: The final publication is available at www.springerlink.com. 30 pages, 34 figure

A combinatorial survey of identities for the double factorial

“…Lattice paths of steps (1, 1), (1, −1), (−1, 1) from (0,0) to (2n, 0) that lie in the first quadrant and do not self-intersect [9].…”

“…Finally, list the edges of T 0 in standard order, that is, in increasing order of their parent vertices (preserving, of course, the order of edges with a common parent vertex), and take the positions of the highlighted edges in this list as the contribution from E to the k-element set X. In the example, T 0 has defining edge list (0, 3), (0, 1), (1,9), (1, 2), (3,8), (3,4), (3,6), (3,5), (4,7) and the highlighted edges-(1, 9), (3,4), (3,5)-are in positions 3,6,8. The net result is the pair (X, T 0 ) with X = {4 V , 10 V , 3 E , 6 E , 8 E } and T 0 as just given.…”

Encoding discrete quantum algebras in a hierarchy of binary words

“…In section 2, we prove that a direct transfer of the axiomatic framework of abstract C*-algebras is possible for a hierarchy of lexicographically ordered (LEX) dictionaries of binary words using the construct of an Inductive Combinatorial Hierarchy (ICH) that was introduced in [44] as a generic toolbox for the study of global maps of automata. In section 3, we observe that the result of the necessary conjugation introduced in the previous section, sends directly to a special subset of words of a Dyck language [45], [46] which is isomorphic with the diagrammatic interpretation of the elements of any TL n algebra. We then examine the closure of any TL n product inside the ICH and we attempt the complete arithmetization of the resulting subset.…”

Generating Functions for Domino Matchings in the $2\times k$ Game of Memory

“…1 Terada [2] and Marsh and Martin [3] have considered Brauer diagrams in the context of combinatorics. The matrices are naturally upper anti-triangular since h + v ≤ k. The sum of the numbers on the anti-diagonal are the Fibonacci numbers, which count the domino tilings of the 2 × k array.…”