## Advanced Studies in Pure Mathematics

- Adv. Stud. Pure Math.
- Representation Theory, Special Functions and Painlevé Equations — RIMS 2015, H. Konno, H. Sakai, J. Shiraishi, T. Suzuki and Y. Yamada, eds. (Tokyo: Mathematical Society of Japan, 2018), 303 - 346

### Rigged Configurations and Catalan, stretched parabolic Kostka numbers and polynomials: Polynomiality, unimodality and log-concavity

#### Abstract

We will look at the Catalan numbers from the *Rigged Configurations* point of view originated [10] 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 the Catalan number $C_n$ as the number of standard Young tableaux of rectangular shape $(n^2)$, or equivalently, as the Kostka number $K_{(n^2),1^{2n}}$, as the starting point of our research. We observe that the rectangular (or multidimensional) Catalan numbers $ C(m,n)$, introduced and studied by P. MacMahon [23], [34], see also [35], can be identified with the corresponding Kostka numbers $K_{(n^m),1^{mn}}$, and therefore can be treated by the 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 [8], [28], [17]. Finally, we give a brief introduction to a rigged configuration version of the Robinson–Schensted–Knuth correspondence.

Another application of the Rigged Configuration technique presented, is a new family of counterexamples to Okounkov's log-concavity conjecture [27].

Finally, we apply Rigged Configurations technique to give a combinatorial proof of the unimodality of the principal specialization of the internal product of Schur functions. In fact we prove a combinatorial (fermionic) formula for generalized $q$-Gaussian polynomials which is a far generalization of the so-called $KOH$-identity [26], as well as it manifests the unimodality property of the $q$-Gaussian polynomials.

#### Article information

**Dates**

Received: 18 August 2015

Revised: 15 December 2015

First available in Project Euclid:
21 September 2018

**Permanent link to this document**

https://projecteuclid.org/
euclid.aspm/1537499430

**Digital Object Identifier**

doi:10.2969/aspm/07610303

**Mathematical Reviews number (MathSciNet)**

MR3837926

**Zentralblatt MATH identifier**

07039307

**Subjects**

Primary: 05E05: Symmetric functions and generalizations 05E10: Combinatorial aspects of representation theory [See also 20C30] 05A19: Combinatorial identities, bijective combinatorics

**Keywords**

Catalan and Narayana numbers stretched Kostka polynomials internal product of Schur functions rigged configurations

#### Citation

Kirillov, Anatol N. Rigged Configurations and Catalan, stretched parabolic Kostka numbers and polynomials: Polynomiality, unimodality and log-concavity. Representation Theory, Special Functions and Painlevé Equations — RIMS 2015, 303--346, Mathematical Society of Japan, Tokyo, Japan, 2018. doi:10.2969/aspm/07610303. https://projecteuclid.org/euclid.aspm/1537499430