Enumeration of trees and one amazing representation of the symmetric group

Abstract. In this paper we present a systematic approach to enumeration of differ-ent classes of trees and their generalizations. The principal idea is nding a bijection between these trees and some classes of Young diagrams or Young tableaux. The latter arise from the remarkable representation of the symmetric group studied by Haiman in connection with diagonal harmonics (see [7]). De ne the vector space V ≃ ⟨xa1 (1): : : xan (n) j 2 Sn; 0 ai i 1; 1 i n⟩. Let the symmetric group Sn act on Vn by the permutation of variables. It is known that dim(Vn) = (n + 1)n1 is equal to the number of labeled trees, and dim(Vn)Sn = 1 n+