On a remarkable partition identity

The starting point of this note is a remarkable partition identity, concerning the parts of the partitions of a fixed natural number and the multiplicities with which these parts occur. This identity is related to the ordinary representation theory of the symmetric group. Our main result is a generalization of this identity, being related to the modular representation theory of the symmetric group. Introduction. The starting point of this note is the remarkable partition identity stated below as Theorem 1. This identity seems to be well-known to combinatorialists, but it is also related to the ordinary representation theory of the symmetric group. We include a proof of Theorem 1 revealing this relationship. Our main result is a generalization of this identity, which is stated below as Theorem 2. While the original identity is related to the ordinary representation theory, its generalization is related to the modular representation theory of the symmetric group. The proof builds on the idea already used in our proof of Theorem 1. The assertion of Theorem 2 gives rise to some numbers called eσ, whose significance, in particular in relation to the modular representation theory of the symmetric group, in most cases is not clear, except for e[1n], which are related to the Cartan determinants of the symmetric group, and for which we give a closed combinatorial formula in Theorem 3. Finally, in Theorem 4, we give a similar formula for certain sums of eσ’s, which are related to alternating groups. Let us first fix the necessary Notation. For n ∈ N0 let Pn denote the set of partitions of n and let pn := |Pn| be its cardinality. For a partition λ ∈ Pn let [λ1, λ2, . . . , λl] be the list of its parts, where λ1 ≥ λ2 ≥ . . . ≥ λl > 0 and ∑l i=1 λi = n. Let lλ = l be its length, and let λi := 0 for all l < i ∈ N. Alternatively, we write λ ∈ Pn as [11 , 22 , . . . , nn ], where aj(λ) = aj := |{1 ≤ i ≤ lλ;λi = j}|. The set Pn can be ordered lexicographically by letting λ >lex μ if and only if for some j ≥ 1 we have λj > μj , while λi = μi for all 1 ≤ i < j. Furthermore, let P n := {λ ∈ Pn;n − lλ even} and P− n := Pn \ P n denote the sets of even and odd partitions, respectively, and let pn := |P± n | be their cardinalities. For n ∈ N let Sn be the symmetric group on n letters. For λ ∈ Pn let χλ ∈ Irr(Sn) denote the corresponding irreducible ordinary character of Sn, see [7, Thm.2.1.11]. For μ ∈ Pn let χλ(μ) := χλ(gμ), where gμ ∈ Sn has cycle type μ. Let Xn = [χλ(μ);λ, μ ∈ Pn] ∈ Zpn×pn be the ordinary character table of Sn, where its rows Date: November 12, 2002. 1991 Mathematics Subject Classification. 05E10, 20C30, 20D60.