## Comments on: C. Krattenthaler, Bijective proofs of the hook formulas for the number of
standard Young tableaux, ordinary and shifted

Comment by the author, November 2, 1998

An elegant hook bijection (perhaps, THE hook bijection) was found by
Pak and Stoyanovskii [1]. It avoids the involution principle. It works
on the basis of jeu de taquin moves which are recorded in a beautiful
manner. However, [1] does not contain a proof that the algorithm works.
A proof is provided in [2]. Using similar ideas, Novelli and Pak [3] claim
to have found a bijection for the shifted hook formula that avoids the
involution principle.

### References

[1] I. M. Pak and A. V. Stoyanovskii, A bijective proof of the hook-length
formula and its analogues, Funct. Anal. Appl. 26 (1992), 216-218;
translated from Funkt. Anal. Priloz. 26 (No. 3) (1992), 80-82.

[2] J.-C. Novelli, I. M. Pak and A. V. Stoyanovsii,
A
direct bijective proof of the hook-length formula,
Discrete Math. Theoret. Computer
Science 1 (1997), 53-67.

[3] J.-C. Novelli and I. M. Pak, in preparation.