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Directed Graphs and the Jacobi-Trudi Identity

Published online by Cambridge University Press:  20 November 2018

I. P. Goulden
I. P. Goulden*
Affiliation:
University of Waterloo, Waterloo, Ontario
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Let |aij|n×n denote the n × n determinant with (i, j)-entry aij, and hk = hk(x1, …, xn) denote the kth-homogeneous symmetric function of x1, …, xn defined by

where the summation is over all m1, …, mn ≧ 0 such that m1 + … + mn = k. We adopt the convention that hk = 0 for k < 0. For integers α1 ≧ α2 … ≧ αn ≧ 0, the Jacobi-Trudi identity (see [6], [7]) states that

In this paper we give a combinatorial proof of an equivalent identity, Theorem 1.1, obtained by moving the denominator on the RHS to the numerator on the LHS.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 37 , Issue 6 , 01 December 1985 , pp. 1201 - 1210
Copyright
Copyright © Canadian Mathematical Society 1985

References

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