A bijection proving orthogonality of the characters of Sn

Suppose ? and β are partitions of n. If ? ? β, a bijection is given between positive pairs of rim hook tableaux of the same shape λ and content β and ?, respectively, and negative pairs of rim hook tableaux of some other shape μ and content β and ?, respectively. If ? = β, the bijection is between positive pairs and either negative pairs or permutations of hooks. The bijection, in the latter case, is a generalization of the Schensted correspondence between pairs of standard tableaux and permutations. If the irreducible characters of S_n are interpreted combinatorially using the Murnaghan-Nakayama formula, these bijections prove

$\text{∑}\text{λ}\text{X}{}^{\mathrm{\lambda }}{}_{\mathrm{\rho }}\text{X}{}^{\mathrm{\lambda }}{}_{\mathrm{\beta }}\text{=δ}{}_{\mathrm{\rho \beta }}\text{1}{}^{\mathrm{j1}}\text{J1!2}{}^{\mathrm{j2}}\text{J2!…}$

where ? = 1^j12^j2….