Four correspondences between graphs and generalized young tableaux

Abstract

Read's method of counting the number of undirected labeled graphs with a prescribed valency at each labeled node implies that the number of different graphs with a given degree sequence (d1, d2, d3...dn) is equal to the number of generalized Young tableaux of a certain shape filled with objects of specification (d1, d2, d3...dn). There are in fact four such results which are applicable to graphs with or without loops and with or without multiple edges. This paper contains four one-one correspondences between the four types of graph and generalized Young tableaux having four different shapes. The correspondences can be considered as combinatorial proofs of four identities of Littlewood. © 1974.