Control theory and opitimization I by M.I. Zelikin, S.A. Vakhrameev

By M.I. Zelikin, S.A. Vakhrameev

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A − n)! (b − n)! (c − n)! (2n)! n=0 a+b b+k = (a + b + c)! a! b! c! (2q + m)! ( − m)! (2q − − m)! =0 2q a+b b+k c+a a+k b+c c+k (−1)k (D) c+a a+k [Foata 1965] [Fjeldstad 1954] si 0 ≤ ≤ 2q (2q + m)! (uv)m (u + v)2q−2m 3 (2q − 2m)! ) m=0 [MacMahon 1915] (2q + m)! 22q−2m 3 (2q − 2m)! ) m=0 3 = (−1)q =0 (3q)! )3 [MacMahon 1915] [Dixon 1890] II. Cas impair : (A ) b+c+1 c+k c+a+1 a+k a+b+1 b+k p = (a + b + c − n + 1)! (a − n)! (b − n)! (c − n)! (n + k)! (n − k + 1)! n=κ lorsque κ = max{−k, k − 1} est major´e par p.

Sq les lettres intervenant dans w, rang´ees par ordre croissant, α(i) la multiplicit´e de si dans w et β(i) = α(1) + · · · + α(i) (avec la convention β(0) = 0). Notons w = x1 · · · xm le r´earrangement croissant α(1) α(q) s1 · · · sq de w et σ la permutation de l’ensemble {1, 2, . . , m} qui satisfasse a ` xi = xσ(i) pour 1 ≤ i ≤ m et qui soit croissante sur chacun des intervalles [β(j − 1) + 1, β(j)] pour 1 ≤ j ≤ q. Par ailleurs, soit (w1 , . . , wq ) la d´ecomposition descendante de w ; on pose γw1 · · · γwp = y1 · · · ym ; on a alors Φ(w) = yσ(1) · · · yσ(m) .

C) Soit c une application de X × X dans un mono¨ıde commutatif Ω, telle que c(x, y) = 0 pour x ≥ y. On a m−1 (14) m−1 c(xi , xi ) = i=1 c(xi , xi+1 ). i=1 On a ∆(w) = Γ(w ). 10, on a (w) = θ(∆(w)) = θ(Γ(w )) = (w ) et ξx,y (w) = nx,y (∆(w)) = nx,y (Γ(w )) = νx,y (w ) pour x < y. Ceci prouve (a) et (b). 10 ; comme on a xm ≥ xm , on a aussi c(xm , xm ) = 0, d’o` u imm´ediatement la formule (14). Soient x = x1 · · · xm un mot et w = x1 · · · xm son r´earrangement croissant. On note ν(w) le nombre des entiers i tels que 1 ≤ i ≤ m et xi < xi et ξ(w) le nombre des entiers i tels que 1 ≤ i ≤ m − 1 et xi < xi+1 .

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