30 research outputs found
The weighted hook length formula
Based on the ideas in [CKP], we introduce the weighted analogue of the
branching rule for the classical hook length formula, and give two proofs of
this result. The first proof is completely bijective, and in a special case
gives a new short combinatorial proof of the hook length formula. Our second
proof is probabilistic, generalizing the (usual) hook walk proof of
Green-Nijenhuis-Wilf, as well as the q-walk of Kerov. Further applications are
also presented.Comment: 14 pages, 4 figure
Hook formulas for skew shapes
International audienceThe celebrated hook-length formula gives a product formula for the number of standard Young tableaux of a straight shape. In 2014, Naruse announced a more general formula for the number of standard Young tableaux of skew shapes as a positive sum over excited diagrams of products of hook-lengths. We give two q-analogues of Naruse's formula for the skew Schur functions and for counting reverse plane partitions of skew shapes. We also apply our results to border strip shapes and their generalizations
Another involution principle-free bijective proof of Stanley's hook-content formula
Another bijective proof of Stanley's hook-content formula for the generating
function for semistandard tableaux of a given shape is given that does not
involve the involution principle of Garsia and Milne. It is the result of a
merge of the modified jeu de taquin idea from the author's previous bijective
proof (``An involution principle-free bijective proof of Stanley's hook-content
formula", Discrete Math. Theoret. Computer Science, to appear) and the
Novelli-Pak-Stoyanovskii bijection (Discrete Math. Theoret. Computer Science 1
(1997), 53-67) for the hook formula for standard Young tableaux of a given
shape. This new algorithm can also be used as an algorithm for the random
generation of tableaux of a given shape with bounded entries. An appropriate
deformation of this algorithm gives an algorithm for the random generation of
plane partitions inside a given box.Comment: 23 pages, AmS-Te