Algebraic Combinatorics by Richard P. Stanley

Author:Richard P. Stanley
Language: eng
Format: epub
Publisher: Springer New York, New York, NY

In the proof of Theorem 8.4 we saw that (see (8.6)). Hence we get

(8.12)

As mentioned after (8.10), the expansion of (D + U) ℓ + 1 in terms of is unique. Hence equating coefficients of on both sides of (8.12) yields the recurrence

(8.13)

It is a routine matter to check that the function satisfies the same recurrence (8.13) as b ij (ℓ), with the same initial condition b 00(0) = 1. From this the proof follows by induction.

From Lemma 8.6 it is easy to prove the following result.

8.7 Theorem

Let ℓ ≥ n and λ ⊢ n, with ℓ − n even. Then

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