Similar constructions for young tableaux and involutions, and their application to shiftable tableaux

which has a straightforward combinatorial proof when an is the number of involutions of [n], must also hold when an is the number of Young tableaux on [n]. In this paper, we first give (Section 3) a combinatorial proof of (1) for Young tableaux which agrees under the Robinson-Schensted correspondence with the proof for involutions. The proof involves a recursive construction which depends in part on Schensted's insertion algorithm [6]. An immediate consequence of this construction is a straightforward proof of the fact that the number of Young tableaux on In] equals the number of involutions on In]. The usual proof requires the use of the result that if a permutation corresponds to the pair of tableaux (P, Q), then its inverse corresponds to the pair (Q, P), but the proof here does not. Another consequence is a bijective proof of a theorem of Schiitzenberger, simpler than in [8], that the number of fixed points of an involution equals the number of columns of odd length in the corresponding tableau. These proofs are discussed in Section 4. We then restrict the construction to shiftable tableaux (definition below) and obtain (in Section 5) a method for listing all shiftable tableaux. We also give (in Section 6) a characterization, in terms of certain decreasing subsequences, of the involutions which correspond to shiftable tableaux under the RobinsonSchensted correspondence, and a recursive description of these involutions.