Abstract
We prove several new instances of the cyclic sieving phenomenon (CSP) on Catalan objects of type \(A\) and type \(B\). Moreover, we refine many of the known instances of the CSP on Catalan objects. For example, we consider triangulations refined by the number of "ears", non-crossing matchings with a fixed number of short edges, and non-crossing configurations with a fixed number of loops and edges.Keywords: Dyck paths, cyclic sieving, Narayana numbers, major index, q-analog.Mathematics Subject Classifications: 05E18, 05A19, 05A30
Highlights
The original inspiration for this paper is a natural interpolation between type A and type B Catalan numbers
Which is known as the nth type B Catalan number, see [Arm09]
Many known instances of the cyclic sieving phenomenon involve a set X whose size is a Catalan number. Once such a CSP triple is obtained, one can ask if X can be partitioned X = jXj in such a way that the group action on X induces a group action on Xj for all j, and, in that case, ask if there is a refinement of the CSP triple in question
Summary
The original inspiration for this paper is a natural interpolation between type A and type B Catalan numbers. For s ∈ {0, 1, n}, the polynomials in (1.2) appear in instances of the cyclic sieving phenomenon. Many known instances of the cyclic sieving phenomenon involve a set X whose size is a Catalan number. Once such a CSP triple is obtained, one can ask if X can be partitioned X = jXj in such a way that the group action on X induces a group action on Xj for all j, and, in that case, ask if there is a refinement of the CSP triple in question. We present a comprehensive (but most likely incomplete) overview of the current state-of-the-art regarding the cyclic sieving phenomenon involving Catalan and Narayana objects of type A and B
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