Oscillating rim hook tableaux and colored matchings

Abstract

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 (λ 0 , λ 1 , . . . , λ 2n ) of Young diagrams starting with the empty shape and ending with the empty shape such that λ i is obtained from λ i−1 by adding an mrim 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 C n C n+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 C n C n+1 .

Crossings and Nestings in Colored Set Partitions

“…In Section 5, we briefly survey several colored variations of these results. Our main finding is that many colored objects−such as matchings [7], permutations [6,37], and tangled diagrams [8]−can be realized as colored set partitions Λ satisfying certain conditions on min(Λ) and max(Λ); see Propositions 5.6, 5.12, and 5.16. The techniques we used to prove Theorem 1.4 can therefore also be used to easily derive the symmetric joint distribution of crossing and nesting numbers in these cases.…”

“…We now describe one generalization of the results in the previous section which will prove useful in enumerating certain classes of colored set partitions. (Chen and Guo investigate another generalization in [7]; the relationship between our statements and those in Chen and Guo's work will be discussed in Section 5.1.) To begin we note the following definition: Definition 3.1.…”

“…In this final section we discuss a few variations of set partitions with natural notions of crossings, nestings, and colorings. Our discussion here is partially expository, surveying results from [6,7,8,9,37]. In each case our noncrossing and nonnesting colored objects are in bijection with certain classes of noncrossing and nonnesting colored set partitions.…”

Dyck Words, Lattice Paths, and Abelian Borders

“…Computer experiments strongly suggest that the following conjectures are true. The integer A191386(n) in Sloane's Online Encyclopedia of Integer Sequences (OEIS) gives the number of ascents of length 1 in all dispersed Dyck paths of length n. A dispersed Dyck path is one or more Dyck paths connected by one or more horizontal steps [8], or equivalently a Motzkin path of length n with no horizontal steps at positive heights. An ascent is a maximal sequence of upward steps.…”

Polynomiality of Certain Average Weights for Oscillating Tableaux

“…Another proof was obtained by Stanley [25,27] in the study of differential posets. The enumerations of various oscillating tableaux with restrictive conditions can be found in [3,4,6,17,16,22]. In 2015, Hopkins and Zhang [13] proved the following result on the average of certain weight function of oscillating tableaux.…”