2011/05/22 by William Y. C. Chen, Chen, William Y. C., Peter L. Guo +1
Mathematics · #05A15 #05A18 #Advanced Combinatorial Mathematics #Advanced Topics in Algebra #Algebraic structures and combinatorial models #Combinatorics (math.CO) #FOS: Mathematics #math.CO #msc:05A15 #msc:05A18
paper · pdf · doi:10.48550/arxiv.1105.4306
15 pages, 16 figures
arxiv created 2011/05/22 · openalex publication_date 2011/05/22 · arxiv updated 2011/05/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Motivated by the question of finding a type B analogue of the bijection between oscillating tableaux and matchings, we find a correspondence between oscillating m-rim hook tableaux and m-colored matchings, where m is a positive integer. An oscillating m-rim hook tableau is defined as a sequence (λ0,λ1,...,λ2n) of Young diagrams starting with the empty shape and ending with the empty shape such that λi is obtained from λi-1 by adding an m-rim hook or by deleting an m-rim hook. Our bijection relies on the generalized Schensted algorithm due to White. An oscillating 2-rim hook tableau is also called an oscillating domino tableau. When we restrict our attention to two column oscillating domino tableaux of length 2n, we are led to a bijection between such tableaux and noncrossing 2-colored matchings on \1, 2,..., 2n\, which are counted by the product CnCn+1 of two consecutive Catalan numbers. A 2-colored matching is noncrossing if there are no two arcs of the same color that are intersecting. We show that oscillating domino tableaux with at most two columns are in one-to-one correspondence with Dyck path packings. A Dyck path packing of length 2n is a pair (D, E), where D is a Dyck path of length 2n, and E is a dispersed Dyck path of length 2n that is weakly covered by D. So we deduce that Dyck path packings of length 2n are counted by CnCn+1.