An Elementary Proof of the Hook Content Formula

Graham H. Hawkes

Published 2013-10-19, updated 2014-10-09Version 2

In this paper we prove the Hook Content Formula (HCF) (and Hook Length Formula (HLF)) using induction. Instead of working with Young tableaux directly, we introduce a vector notation (sequences of these vectors represent ("single-vote") ballot sequences in the case of SYT, and "multi-vote" ballot sequences--where the voter may choose any number of candidates--in the case of SSYT) to aid in the inductive argument. Next, we establish an identity which allows us to prove a formula that counts multi-vote ballot sequences. We demonstrate that, in the non-degenerate case (when this formula counts SSYTs) it coincides with the HCF. To do the latter, we borrow parts of a technique outlined by Wilson and Van Lint in their proof of the HLF. We then establish an identity, which is really a special case of the equation mentioned above, and show that the HLF follows from it. (Wilson and Van Lint prove this identity directly, and use it to prove the HLF in a similar manner.) We then note the appearance of an expression resembling the Weyl dimension formula and conclude with a combinatorial result.