KAIST Discrete Math Seminar

The Catalan number is perhaps the most frequently occurred number in combinatorics. Richard Stanley has collected more than 170 combinatorial objects counted by the Catalan number. Noncrossing partition, which has received great attention recently, is one of these, so called, Catalan objects. Noncrossing partitions are generalized to each finite Coxeter group. In this talk, we will interpret noncrossing partitions of type B in terms of noncrossing partitions of type A. As applications, we can prove interesting properties of noncrossing partitions of type B.

We prove combinatorially that the largest power of 2 in the number of involutions of length n is equal to [n/2]-2[n/4] +[(n+1)/4]. We show that the smallest period of the sequence of odd factors in the number of involutions modulo 2^s is 2^s+1 for s>2. We also consider the largest power of 2 in the number of even and odd involutions.