Generalized Binomial Coefficients for Molecular Species

✍ Scribed by Pierre Auger; Gilbert Labelle; Pierre Leroux

• Publisher: Elsevier Science
• Year: 2000
• Tongue: English
• Weight: 351 KB
• Volume: 91
• Category: Article
• ISSN: 0097-3165
• DOI: 10.1006/jcta.2000.3088

✦ Synopsis

Let ! be a complex variable. We associate a polynomial in !, denoted ( M N ) ! , to any two molecular species M=M(X) and N=N(X) by means of a binomial-type expansion of the form

In the special case M(X)=X m , the species of linear orders of length m, the above formula reduces to the classical binomial expansion

When !=1, a M(1+X)-structure can be interpreted as a partially labelled M-structure and ( M N ) 1 is a nonnegative integer, denoted ( M N ) for simplicity. We develop some basic properties of these ``generalized binomial coefficients'' and apply them to study solutions, 8, of combinatorial equations of the form M(8)=9 in the context of C-species, M being molecular and 9 being a given C-species. This generalizes the study of symmetric square roots (where M=E 2 , the species of 2-element sets) initiated by P.