Restricted Sum Formula of Alternating Euler Sums

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In this paper we study restricted sum formulas involving alternating Euler sums which are defined by ζ(s₁,...,sd;ε₁,...,εd)=∑ₙ₁>...>ₙd≥ ₁ε₁ⁿ¹... εdⁿᵈn₁ˢ¹... ndˢᵈ, for all positive integers s₁,...,sd and ε₁=± 1,..., εd=± 1 with (s₁,ε₁) unequal (1,1). We call w=s₁+...+sd the weight and d the depth. When εⱼ=-1 we say the jth component is alternating. We first consider Euler sums of the following special type: ξ(2s₁,...,2sd)=ζ(2s₁,...,2sd;(-1)ˢ¹,...,(-1)sd). For d≤ n, let Ξ(2n,d) be the sum of all ξ(2s₁,..., 2sd) of fixed weight 2n and depth d. We derive a formula for Ξ(2n,d) using the theory of symmetric functions established by Hoffman recently. We also consider restricted sum formulas of Euler sums with fixed weight 2n, depth d and fixed number αof alternating components at even arguments. When α=1 or α=d we can determine precisely the restricted sum formulas. For other αwe only treat the cases d<5 completely since the symmetric function theory becomes more and more unwieldy to work with when αmoves closer to d/2.