The spectrum of a Hermitian matrix sum

An announcement by Lidskii, (B.V. Lidskii, Functional Anal. Appl. 10 (1982) 76–77 (Russian), 139–140 (English)), claimed to establish the explicit description of the spectrum of a Hermitian matrix sum in the form conjectured by Horn (A. Horn, Pacific J. Math. 16 (1962) 225–241), but no supporting proof has been published. This paper begins an analysis of the claim, and is the first step towards bridging the distance between the elementary methods of Horn's (1962) paper and the partial solution of the same problem using noncommutative harmonic analysis by Dooley et al. (I. Dooley, R. Repka, N. Wildberger, Lin. Mult. Alg. 36 (1993) 79–102). The methods of Horn/Lidskii lead to combinatorial issues of independent interest.