Many invertible actions τ on a set S of combinatorial objects, along with a natural statistic f on S, exhibit the following property which we dub homomesy: the average of f over each τ -orbit in S is the same as the average of f over the whole set S. This phenomenon was first noticed by Panyushev in 2007 in the context of the rowmotion action on the set of antichains of a root poset; Armstrong,...

The Gelfand–Tsetlin graph is an infinite graded graph that encodes branching of irreducible characters of the unitary groups. The boundary of the Gelfand–Tsetlin graph has at least three incarnations — as a discrete potential theory boundary, as the set of finite indecomposable characters of the infinitedimensional unitary group, and as the set of doubly infinite totally positive sequences. An ...

Let the sign of a standard Young tableau be the sign of the permutation you get by reading it row by row from left to right, like a book. A conjecture by Richard Stanley says that the sum of the signs of all SYTs with n squares is 2. We present a stronger theorem with a purely combinatorial proof using the Robinson-Schensted correspondence and a new concept called chess tableaux. We also prove ...

Let the sign of a standard Young tableau (SYT) be permutation obtained by reading entries row from left to right, starting top row. By sign-reversing involution on tableaux in terms lattice paths, we obtain sign-balance $n$-cell SYTs all shapes with at most three rows. For skew shapes, partial results enumeration $2n$-cell rows, under parity condition part.

The jeu de taquin process produced a standard Young tableau from a skew standard Young tableau by shifting its entries to the northwest. We generalize this process to posets: certain partial numberings of any poset are shifted upward. A poset is said to have the jeu de taquin property if the numberings resulting from this process do not depend upon certain choices made during the process. Young...

The irreducible representations of the symmetric group Sn are parameterized by combinatorial objects called Young diagrams, or shapes. A given irreducible representation has a basis indexed by Young tableaux of that shape. In fact, this basis consists of weight vectors (simultaneous eigenvectors) for a commutative subalgebra F[X] of the group algebra FSn. The double affine Hecke algebra (DAHA) ...

We express general conjectures for explicit forms of P-recurrences for the number of Young standard tableaux of height bounded by h. These recurrences are compatible with known results and Regev's asymptotic evaluations. Le but de cette note est essentiellement de pr esenter des conjectures ainsi que des indications sur les raisons qui nous ont port ees a les enoncer. Nos conjectures concernent...

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”, Di...