Generalization of the FRT Hook Length Formula for Counting Standard Young Tableaux

Fix a shape lambda and let P be the corresponding poset (which has one element for each box in lambda). It is not hard to see that the recipe for the h_y's given in the previous page produces the same hooks that the recipe of Frame-Robinson-Thrall produces. Each standard Young tableau on lambda is an order extension of P to a total order.

Let P be any poset. Then the number e(P) of extensions of P to a total order can be recovered from Stanley's P-partition generating function for P, from the leading coefficient of the polynomial for the coefficient of x^k as k goes to infinity.

If P is a d-complete poset with n elements, then the number of order extensions of P to a total order is just n! divided by the product of the hook lengths h_y over the elements y of P.

This is the Corollary stated on our homepage: Theorem 2 contains a generalization of the FRT formula to any d-complete poset.

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