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doi:10.1016/S0012-365X(97)00110-6 
Copyright © 1998 Published by Elsevier B.V.
Left-inversion of combinatorial sums
Received 18 July 1995;
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Abstract
The inversion of combinatorial sums is a fundamental problem in algebraic combinatorics. Some combinatorial sums, such as an = Σkdn,kbk, cannot be inverted in terms of the orthogonality relation because the infinite, lower triangular array P = {dn,k}'s diagonal elements are equal to zero (except d0,0). Despite this, we can find a left-inverse ̄P such that PP̄ = I and therefore are able to left-invert the original combinatorial sum, and thus obtain bn = Σkd̄n,kak.
Résumé
L'inversion des sommes combinatoires est un problème fondamental dans l'algèbre combinatoire. Certaines sommes combinatoires, par exemple an = Σkdn,kbk, ne peuvent pas être inverties selon la rélation d'orthogonalité, parce que les éléments sur la diagonale de la matrice triangulaire inférieure P = {dn,k} sont nuls (sauf d0,0). Malgré cela, on peut bien souvent définir une matrice left-inverse P̄ telle que PP̄ = I et, par conséquent, on peut left-invertir la somme combinatoire d'origine, en obtenant bn = Σkd̄n,kak.
