Abstract. We present results on the enumeration of crossings and nestings for matchings and set partitions. Using a bijection between partitions and vacillating tableaux, we show that if we fix the sets of minimal block elements and maximal block elements, the crossing number and the nesting number of partitions have a symmetric joint distribution. It follows that the crossing numbers and the nesting numbers are distributed symmetrically over all partitions of [n], as well as over all matchings on [2n]. As a corollary, the number of knoncrossing partitions is equal to the number of k-nonnesting partitions. The same is also true for matchings. An application is given to the enumeration of matchings with no k-crossing (or with no k-nesting).

In this paper, we present a reduction algorithm which transforms m-regular partitions of [n] = {1, 2, . . . , n} to (m − 1)-regular partitions of [n − 1]. We show that this algorithm preserves the noncrossing property. This yields a simple explanation of an identity due to Simion-Ullman and Klazar in connection with enumeration problems on noncrossing partitions and RNA secondary structures. For ordinary noncrossing partitions, the reduction algorithm leads to a representation of noncrossing partitions in terms of independent arcs and loops, as well as an identity of Simion and Ullman which expresses the Narayana numbers in terms of the Catalan numbers.

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Riordan paths are Motzkin paths without horizontal steps on the x-axis. We establish a correspondence between Riordan paths and (321, 3142)-avoiding derangements. We also present a combinatorial proof of a recurrence relation for the Riordan numbers in the spirit of the Foata-Zeilberger proof of a recurrence relation on the Schröder numbers.

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Let A k be the set of permutations in the symmetric group S k with prefix 12. This paper concerns the enumeration of involutions which avoid the set of patterns A k . We present a bijection between symmetric Schröder paths of length 2n and involutions of length n + 1 avoiding A4. Statistics such as the number of right-to-left maxima and fixed points of the involution correspond to the number of steps in the symmetric Schröder path of a particular type. For each k ≥ 3 we determine the generating function for the number of involutions avoiding the subsequences in A k , according to length, first entry and number of fixed points.A k = {π 1 π 2 . . . π k ∈ S k | π 1 = 1, π 2 = 2}.2000 Mathematics Subject Classification. Primary: 05A05, Secondary: 05A15.