R P Stanley Enumerative combinatorics Solution de l'exercice

Description

R. P. Stanley, Enumerative combinatorics. — Solution de l'exercice 5.37 REDACTION & CORRESPONDANCE : MATTHIEU DENEUFCHATEL, CIP 09-03-2010 13:37 Contents 1 Preamble 1 2 Solution of a). 2 3 Solution of b). 5 3.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.2 Solution of the first part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3.3 Solution of the second part . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 4 Solution of c). 10 5 Solution of d). 10 6 Solution of e). 11 1 Preamble As this exercise uses mainly exponentials, the following result has to be stated. Proposition 1.1 Let A be a Q-algebra and, for a ? A and k ? N define ( a k ) := a(a? 1) · · · (a? k + 1) k! (1) Then, in A[[X]] one has ea ln(1+X) = ∑ k≥0 ( a k ) Xk .

• ?ay ?10


• investigate also3


• following equa- tion


• x? pn


• galois differential

Informations

• Publié par: profil-urra-2012
• Nombre de lectures: 16
• Langue: English

Extrait

R. P. Stanley, Enumerative combinatorics. — Solution de l’exercice 5.37

R  E DAC T IO N & C O R R E S P O N DA N C E : M AT T H IE U D E N E U F C H AT E L , CIP

Contents

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Preamble

Solution of a).

09-03-2010 13:37

Solution of b). 3.1 Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Solution of the first part . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Solution of the second part . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Solution of c).

Solution of d).

Solution of e).

Preamble

As this exercise uses mainly exponentials, the following result has to be stated.

Proposition 1.1LetAbe aQ-algebra and, fora∈ Aandk∈ N define   a a(a−1)∙ ∙ ∙(a−k+ 1) := k k !

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(1)

Then, inA[[X]]one has   X a aln(1+X)k e=X .(2) k k≥0 a This last quantity will be denoted(1 +X). Of course, due to (2), one has for commuting a+bb a a, b∈ A,(1 +X(1 +) = X) (1 +X).

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