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By Carl B. Boyer, Uta C. Merzbach, Isaac Asimov

Boyer and Merzbach distill millions of years of arithmetic into this interesting chronicle. From the Greeks to Godel, the maths is fabulous; the forged of characters is unique; the ebb and movement of rules is far and wide glaring. And, whereas tracing the advance of eu arithmetic, the authors don't put out of your mind the contributions of chinese language, Indian, and Arabic civilizations. surely, this is—and will lengthy remain—a vintage one-volume heritage of arithmetic and mathematicians who create it.

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R . 5. — Pour r ≥ 1 on a : rA(t, u) = (r − 1)! 1A(t, u) . The D´emonstration. — Pour r = 1, il n’y a rien ` a prouver. Supposons r ≥ 2 et prenons pour µ le morphisme prolongeant l’application σ → θ∆Dσ = t|∆Dσ| au mono¨ıde S+ . Si l’on a σ ∈ Sn+r−1 et σw = g1 i1 g2 i2 . . gr−1 ir−1 gr 36 ´ ERATRICES ´ CHAPITRE IV : FONCTIONS GEN avec {i1 , i2 , . . , ir−1 } = [r − 1], il vient |∆Dgj | = |∆Dωgj | (j ∈ [r]), puisque ω est un morphisme injectif. D’autre part, puisque le mot g1 g2 , . . gr contient toutes les lettres du mot σw sup´erieures ou ´egales ` a r, on a r−1 r−1 |∆ ∆Dσ| = j |∆Dωgj |, d’o` u µ r σ = θ∆ ∆Dσ.

Pour r = 1, il n’y a rien ` a prouver. Supposons r ≥ 2 et prenons pour µ le morphisme prolongeant l’application σ → θ∆Dσ = t|∆Dσ| au mono¨ıde S+ . Si l’on a σ ∈ Sn+r−1 et σw = g1 i1 g2 i2 . . gr−1 ir−1 gr 36 ´ ERATRICES ´ CHAPITRE IV : FONCTIONS GEN avec {i1 , i2 , . . , ir−1 } = [r − 1], il vient |∆Dgj | = |∆Dωgj | (j ∈ [r]), puisque ω est un morphisme injectif. D’autre part, puisque le mot g1 g2 , . . gr contient toutes les lettres du mot σw sup´erieures ou ´egales ` a r, on a r−1 r−1 |∆ ∆Dσ| = j |∆Dωgj |, d’o` u µ r σ = θ∆ ∆Dσ.

Si l’on a σ ∈ Sn+r−1 et σw = g1 i1 g2 i2 . . gr−1 ir−1 gr 36 ´ ERATRICES ´ CHAPITRE IV : FONCTIONS GEN avec {i1 , i2 , . . , ir−1 } = [r − 1], il vient |∆Dgj | = |∆Dωgj | (j ∈ [r]), puisque ω est un morphisme injectif. D’autre part, puisque le mot g1 g2 , . . gr contient toutes les lettres du mot σw sup´erieures ou ´egales ` a r, on a r−1 r−1 |∆ ∆Dσ| = j |∆Dωgj |, d’o` u µ r σ = θ∆ ∆Dσ. 2, µ r {Sn+r−1 } = rAn+r−1 (t). 5 r´esulte de l’identit´e (11). 3. Autres interpr´ etations des polynˆ omes eul´ eriens Les techniques du chapitre pr´ec´edent pourraient ˆetre appliqu´ees ` a d’autres probl`emes d’´enum´eration.

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