RIMS - 1824 Rigged Configurations and Catalan , Stretched Parabolic Kostka

We will look at the Catalan numbers from the Rigged Configurations point of view originated [9] from an combinatorial analysis of the Bethe Ansatz Equations associated with the higher spin anisotropic Heisenberg models . Our strategy is to take a combinatorial interpretation of Catalan numbers Cn as the number of standard Young tableaux of rectangular shape (n2), or equivalently, as the Kostka number K(n2),12n , as the starting point of research. We observe that the rectangular (or multidimensional) Catalan numbers C(m,n) introduced and studied by P. MacMahon [21], [30], see also [31], can be identified with the Kostka number K(nm),1mn , and therefore can be treated by Rigged Configurations technique. Based on this technique we study the stretched Kostka numbers and polynomials, and give a proof of “ a strong rationality “ of the stretched Kostka polynomials. This result implies a polynomiality property of the stretched Kostka and stretched Littlewood–Richardson coefficients [7], [26], [16]. Another application of the Rigged Configuration technique presented, is a new family of counterexamples to Okounkov’s log-concavity conjecture [25]. Finally, we apply Rigged Configurations technique to give a combinatorial prove of the unimodality of the principal specialization of the internal product of Schur functions. In fact we prove a combinatorial formula for generalized q-Gaussian polynomials which is a far generalization of the so-called KOH-identity [24], as well as it manifests the unimodality property of the q-Gaussian polynomials. 2000 Mathematics Subject Classifications: 05E05, 05E10, 05A19.