a b s t r a c tMaking use of a combinatorial approach, we prove two refined major-balance identities on the 321-avoiding involutions in S n , respecting the number of fixed points and the number of descents, respectively. The former one is proved in terms of ordered trees whose non-root nodes have exactly two children, and the latter one is proved in terms of lattice paths within a ⌊ n 2 ⌋ × ⌈ n 2 ⌉ rectangle.

A permutation σ ∈ Sn is simsun if for all k, the subword of σ restricted to {1, . . . , k} does not have three consecutive decreasing elements. The permutation σ is double simsun if both σ and σ −1 are simsun. In this paper we present a new bijection between simsun permutations and increasing 1-2 trees, and show a number of interesting consequences of this bijection in the enumeration of pattern-avoiding simsun and double simsun permutations. We also enumerate the double simsun permutations that avoid each pattern of length three.

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