J Algebr Comb (2018) 47:241–300
· Nathan Reading
Received: 23 May 2016 / Accepted: 22 June 2017 / Published online: 10 August 2017
© Springer Science+Business Media, LLC 2017
Abstract We pose counting problems related to the various settings for Coxeter-
Catalan combinatorics (noncrossing, nonnesting, clusters, Cambrian). Each problem is
to count “twin”pairsof objects from a corresponding problem in Coxeter-Catalan com-
binatorics. We show that the problems all have the same answer, and, for a given ﬁnite
Coxeter group W , we call the common solution to these problems the W-biCatalan
number. We compute the W-biCatalan number for all W and take the ﬁrst steps in the
study of Coxeter-biCatalan combinatorics.
Keywords Alternating arc diagram · Coxeter-Catalan combinatorics · Doubled root
poset · Twin clusters · Twin noncrossing partitions · Twin nonnesting partitions ·
Twin sortable elements
This paper considers enumeration problems closely related to Coxeter-Catalan combi-
natorics. (For background on Coxeter-Catalan combinatorics, see for example [5,18]).
Each enumeration problem can be thought of as counting pairs of “twin” Coxeter-
Catalan objects—twin sortable elements or twin nonnesting partitions, etc. Many of
the terms used in this introductory section are new to this paper and will be explained
in Sect. 2.
Emily Barnard was supported in part by NSF Grants DMS-0943855, DMS-1101568, and DMS-1500949.
Nathan Reading was supported in part by NSF Grants DMS-1101568 and DMS-1500949.
Department of Mathematics, North Carolina State University, Raleigh, NC, USA