We present an extensive study of the Eulerian distribution on the set of self evacuated involutions, namely, involutions corresponding to standard Young tableaux that are fixed under the Schützenberger map. We find some combinatorial properties for the generating polynomial of such distribution, together with an explicit formula for its coefficients. Afterwards, we carry out an analogous study ...

Many enumerative problems can be expressed using the scalar product of symmetric functions. As we shall see, we can set up expressions for the generating functions of objects which possess a certain kind of regularity as a scalar product. Two examples of this are k-regular graphs (graphs in which each vertex is of degree k), and secondly we have a class of semi-standard Young tableaux in which ...

We prove an equation conjectured by Okada regarding hook-lengths of partitions, namely that 1 n! ∑ λ⊢n f λ ∑ u∈λ r ∏ i=1 (hu − i) = 1 2(r + 1)2 ( 2r r )( 2r + 2 r + 1 ) r ∏ j=0 (n− j), where fλ is the number of standard Young tableaux of shape λ and hu is the hook length of the square u of λ. We also obtain other similar formulas.

We describe one-dimensional central measures on numberings (tableaux) of ideals partially ordered sets (posets). As the main example, we study poset $$\mathbb{Z}_+^d$$ and graph its finite ideals, multidimensional Young tableaux; for $$d=2$$ , this is ordinary graph. The are stratified by dimension; in paper give a complete description stratum prove that every ergodic measure uniquely determine...

The Schur function indexed by a partition λ with at most n parts is the sum of the weight monomials for the Young tableaux of shape λ. Let π be an npermutation. We give two descriptions of the tableaux that contribute their monomials to the key polynomial indexed by π and λ. (These polynomials are the characters of the Demazure modules for GL(n).) The “atom” indexed by π is the sum of weight mo...

The arctic circle theorem of Jockusch, Propp, and Shor asserts that uniformly random domino tilings of an Aztec diamond of high order are frozen with asymptotically high probability outside the “arctic circle” inscribed within the diamond. A similar arctic circle phenomenon has been observed in the limiting behavior of random square Young tableaux. In this paper, we show that random domino tili...

The Tamari lattice Tn was originally defined on bracketings of a set of n+1 objects, with a cover relation based on the associativity rule in one direction. Although in several related lattices, the number of maximal chains is known, quoting Knuth, “The enumeration of such paths in Tamari lattices remains mysterious.” The lengths of maximal chains vary over a great range. In this paper, we focu...

We identify a subalgebra Ĥ + n of the extended affine Hecke algebra Ĥn of type A. The subalgebra Ĥ + n is a u-analogue of the monoid algebra of Sn⋉Z n ≥0 and inherits a canonical basis from that of Ĥn. We show that its left cells are naturally labeled by tableaux filled with positive integer entries having distinct residues mod n, which we term positive affine tableaux (PAT). We then exhibit a ...