M. BERGERSON, A. MILLER, A. PLIML, V. REINER, P. SHEARER, D. STANTON, AND N. SWITALA
` ´ 2n−r−1 Abstract.It is shown that there arenoncrossing partitions of an n−r ` ´`´ n n−r−1 nset together with a distinguished block of sizerof these, and k−1k−2 havekrapnititwsnoohtiregaltsuB´ofaoonnecrossing.lbinizaleren,gksoc Furthermore, when one evaluates naturalqanalogues of these formulae forq th annroot of unity of orderd, one obtains the number of such objects having dfold rotational symmetry.
Given a partitionπof the set [n] :={1,2, . . . , n}, acrossinginπis a quadruple of integers (a, b, c, d) with 1≤a < b < c < d≤nfor whicha, care together in a block, andb, dIt is wellknown [10, Exericsesare together in a different block. 6.19(pp)],[4] that the number ofnoncrossing partitionsof [n] (that is, those with 1 2n no crossings) is the Catalan numberCnand the number of noncrossing= , n+1n 1n n partitions of [n] intokblocks is the Narayana number. n k−1k OurstartingpointisthemorerecentobservationofB´ona[2,Theorem1]that the number of partitions of [n] havingexactly onecrossing has the even simpler 2n−5 formula.B´ona’sproofutilizesthefactthatCnis also wellknown to count n−4 triangulations of a convex (nthis allows him to biject 1crossing partitions+ 2)gon; of [n] to dissections of anngon that use exactlyn−4 diagonals. The proof is then 1n+d−1n−3 completed by pluggingd=n−of Kirkman4 into the formula d+1d d (first proven by Cayley; see [7]) for the number of dissections of anngon usingd diagonals. ThegoalhereistogeneralizeBo´na’sresulttocount1crossingpartitionsbytheir number of blocks, and also to examine a naturalqanalogue with regard to thecyclic sieving phenomenonshown in [8] for certainqCatalan andqNarayana numbers. The crux is the observation that 1crossing partitions of [n] biject naturally with noncrossing partitions of [n] having a distinguished 4element block:replace the crossing pair of blocks{a, c},{b, d}with a single distinguished block{a, b, c, d}. Thus one should count the following more general objects.
Definition 1.Anrblocked noncrossing partition of[n] is a pair (π, B) of a non crossing partitionπtogether with a distinguishedrelement blockBofπ.
Note that the notion of a crossing in a partition is invariant under cyclic rotations i7→i+ 1modnof the set [nthe cyclic group]. ConsequentlyC=Znacts on the
Date: August 2006. Key words and phrases.noncrossing partition, cyclic sieving phenonomenon. This work was the result of an REU at the University of Minnesota School of Mathematics in Summer 2006, mentored by V. Reiner and D. Stanton, and supported by NSF grants DMS0601010 and DMS0503660.The authors also thank D. Armstrong for helpful conversations. 1