Bounded Littlewood identities with fixed number of odd rows or odd columns

TL;DR

This work advances the theory of bounded Littlewood identities by refining column-bounded sums of Schur functions to fix the number of odd-length columns, yielding determinant expressions tied to $f_r(oldsymbol{x})$ and a latent Pfaffian structure. It connects these refined identities to nearly rectangular so-character identities, offering both determinant and skewing-operator formulations, and situates them within nonintersecting lattice-path combinatorics via the Lindström–Gessel–Viennot framework. The paper also derives new formulas for the number of standard Young tableaux of bounded width, and provides rich combinatorial interpretations in terms of up-down tableaux and vacillating/lattice-walk models, unifying symmetric-function techniques, representation theory of classical groups, and path-enumeration methods. Together, these results extend classical Littlewood identities, illuminate connections to orthogonal-group characters, and yield new enumerative formulas for restricted-shape SYTs with concrete combinatorial realizations. The methods—minor-summation formulas, Pfaffian-to-determinant reductions, and skewing-operator formalism—enhance the toolkit for studying bounded symmetric-function identities and their tableau- and path-theoretic incarnations.

Abstract

A Littlewood identity is an identity equating a sum of Schur functions with an infinite product. A bounded Littlewood identity is one where the sum is taken over the partitions with a bounded number of rows or columns. The price to pay is that the infinite product has to be replaced by a determinant. The focus of this article is on refinements of such bounded Littlewood identities where one also prescribes the number of odd-length rows or columns of the partitions. Goulden [{\it Discrete Math.} {\bf99} (1992), 69--77] had given such a refinement in which the number of columns is bounded and the number of odd-length rows is prescribed. We provide refinements where the number of columns is bounded and the number of odd-length columns is prescribed. Furthermore, we present new formulations of such bounded Littlewood identities involving skewing operators. As corollaries we obtain non-standard formulas for numbers of standard Young tableaux with restricted shapes as above. In the last part of the article we discuss combinatorial interpretations of such identities in terms of up-down tableaux. As corollaries, we obtain identities between numbers of standard Young tableaux and numbers of (marked) vacillating tableaux.

Figures (7)

• Figure 1:
• Figure 2: An odd-branch point $u$ (left) and an even-branch point $v$ (right).
• Figure 3: Configurations for the involution in Lemma \ref{['lem:1']}.
• Figure 4: Suppose that the black path $P_r=P'_rP"_r$ has a marking $j\in M_r$. It intersects with the red path $P_s=P'_sP"_s$ at $u$ but this cannot be canceled because the configuration (on the right) obtained by exchanging the two subpaths after $u$ is not a valid configuration due to the marking $j\in M_r$.
• Figure 5: An illustration of the involution in Lemma \ref{['lem:5']}.

Theorems & Definitions (70)

• Theorem 1.1: Two bounded Littlewood identities
• Theorem 1.2: Goulden1992
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7
• Theorem 1.8: KLO
• Theorem 1.9
• Theorem 1.10: Zeilberger Zeilberger_lazy