Catalan numbers are generalized to type B: https://oeis.org/A000984.
Are there some references about Catalan numbers of type C, D? Thank you very much.
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asked Apr 26, 2017 at 16:59
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$\begingroup$ You may find this question and its references interesting mathoverflow.net/questions/17197/… $\endgroup$– j.c.Commented Apr 26, 2017 at 17:05
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1$\begingroup$ see arxiv.org/abs/hep-th/0111053 $\endgroup$– F. C.Commented Apr 26, 2017 at 17:32
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1 Answer
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The Catalan numbers you get in types other than A depend on which interpretation of the Catalan numbers you are generalizing. There are at least two possibilities: the number of anti-chains in the root poset (these are sometimes called non-nesting partitions) gives the so called Coxeter-Catalan number; these are well-studied, see e.g. "A uniform bijection between nonnesting and noncrossing partitions" by Armstrong, Stump, and Thomas. The number of Stembridge's "fully commutative" elements (in type A, this is the same as 321-avoiding permutations) of the Coxeter group is another generalization which gives different numbers, see "The enumeration of fully commutative elements of coxeter Groups" by Stembridge for explicit generating functions in all types.
answered Apr 26, 2017 at 17:31
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$\begingroup$ thank you very much. Are there some generalized Catalan number of type B such that the seqnece of the numbers is $3,9,29,97,333$ for $n=2,3,4,5,6$? It seems that the two generalization you gave do not give this sequence. $\endgroup$ Commented Oct 19, 2018 at 8:06
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$\begingroup$ Your sequence appears in OEIS. The generating function looks similar to the one for Catalan numbers, and according to one of the comments it is a "Catalan transform" of the Fibonacci numbers. $\endgroup$ Commented Oct 19, 2018 at 11:43