A combinatorial application of matrix Riccati equations and their q-analogue

✍ Scribed by G.B. Collins; J.P. Goulden; D.M. Jackson; A.M. Nierstrasz

• Publisher: Elsevier Science
• Year: 1981
• Tongue: English
• Weight: 737 KB
• Volume: 36
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/s0012-365x(81)80012-x

✦ Synopsis

The generating functions for a large class of combinatorial problems involving the enumeration of permutations may be expressed as solutions to matrix Riccati equations. We show that the generating functions for the permutation problem in which the number of inversions is also preserved form a system of matrix Riccati equations in which the differential operator is the Eulerian differential operator.

We obtain the classical result of MacMahon concerning permutations.