A Bijective Proof of the Hook Formula for the Number of Column Strict Tableaux with Bounded Entries

A formula for the generating function for the number of column strict tableaux of a fixed shape λ with entries bounded by some positive number a was proved by Littlewood and given an expression in terms of the hook and content numbers of the Ferrers diagram of shape A by Stanley. We give a bijective proof of this formula and in the process give a combinatorial explanation for the appearance of the hook and content numbers in the formula

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

Plane overpartitions and cylindric partitions

Generating functions for plane overpartitions are obtained using various methods such as nonintersecting paths, RSK type algorithms and symmetric functions. We extend some of the generating functions to cylindric partitions. Also, we show that plane overpartitions correspond to certain domino tilings and we give some basic properties of this correspondence.Comment: 42 pages, 11 figures, corrected typos, revised parts, figures redrawn, results unchange

Enumeration of Standard Young Tableaux

A survey paper, to appear as a chapter in a forthcoming Handbook on Enumeration.Comment: 65 pages, small correction