Note On 1-Crossing Partitions

of Kirkman (first proven by Cayley; see [7]) for the number of dissections of an n-gon using d diagonals. The goal here is to generalize Bóna’s result to count 1-crossing partitions by their number of blocks, and also to examine a natural q-analogue with regard to the cyclic sieving phenomenon shown in [8] for certain q-Catalan and q-Narayana numbers. The crux is the observation that 1-crossing partitions of [n] biject naturally with noncrossing partitions of [n] having a distinguished 4-element block: replace the crossing pair of blocks {a, c}, {b, d} with a single distinguished root block {a, b, c, d}. An example is shown in Figure (a), where the 1-crossing partition of [18] having blocks {1, 10}, {2, 3, 4, 5}, {6, 15}, {7, 8}, {9}, {11, 12, 13, 14}, {16, 17}, {18} is shown in its circular representation, with the two blocks {1, 10}, {6, 15} responsible for the unique crossing pair. Figure (a) shows the corresponding noncrossing partition of [n] = [18] with distinguished 4-element root block {a, b, c, d} = {1, 6, 10, 15} that replaced the crossing pair of blocks. Thus one is motivated to count the following more general objects.