The number of standard Young tableaux of a fixed shape is famously given by the hook-length formula due to Frame, Robinson and Thrall. A bijective proof of Novelli, Pak and Stoyanovskii relies on a sorting algorithm akin to jeu-de-taquin which transforms an arbitrary filling of a partition into a standard Young tableau by exchanging adjacent entries. Recently, Krattenthaler and Müller defined t...

The sl3 spider is a diagrammatic category used to study the representation theory of the quantum group Uq(sl3). The morphisms in this category are generated by a basis of non-elliptic webs. KhovanovKuperberg observed that non-elliptic webs are indexed by semistandard Young tableaux. They establish this bijection via a recursive growth algorithm. Recently, Tymoczko gave a simple version of this ...

Young’s lattice is the lattice of partitions of integers, ordered by inclusion of diagrams. Standard Young tableaux can be represented as paths in Young’s lattice that go up by one square at each step, and more general paths in Young’s lattice correspond to more general kinds of tableaux. Using the theory of symmetric functions, in particular Pieri’s rule for multiplying a Schur function by a c...

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...