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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 (x 1, …, xn ) denote the kth-homogeneous symmetric function of x 1, …, xn defined by

where the summation is over all m 1, …, mn ≧ 0 such that m 1 + … + mn = k. We adopt the convention that hk = 0 for k < 0. For integers α 1 ≧ α 2 … ≧ 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.

Information

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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