pith. sign in

arxiv: 2510.11054 · v2 · submitted 2025-10-13 · 🧮 math.CO

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

JiSun Huh , Jang Soo Kim , Christian Krattenthaler , Soichi Okada This is my paper

Pith reviewed 2026-05-18 08:10 UTC · model grok-4.3

classification 🧮 math.CO
keywords bounded Littlewood identitiesodd-length columnsskewing operatorsstandard Young tableauxvacillating tableauxSchur functionsdeterminant formulasup-down tableaux
0
0 comments X

The pith

Refinements of bounded Littlewood identities prescribe the number of odd-length columns and use skewing operators for determinant formulas.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper refines bounded Littlewood identities, which equate sums of Schur functions over partitions with bounded rows or columns to determinant expressions. Goulden previously refined the case with bounded columns and prescribed odd rows. The authors provide the version with prescribed odd columns instead. They also give new formulations using skewing operators applied to the classical product. These yield formulas for counting standard Young tableaux of restricted shapes and combinatorial interpretations involving vacillating tableaux.

Core claim

We obtain refinements of bounded Littlewood identities in which the number of columns is bounded and the number of odd-length columns is fixed, expressed as determinants arising from skewing operators applied to the unbounded Littlewood product. As consequences, we derive non-standard expressions for the number of standard Young tableaux with shapes having bounded columns and fixed odd columns, along with identities relating these to counts of marked vacillating tableaux.

What carries the argument

Skewing operators applied to the classical bounded Littlewood product to produce determinant representations for the generating functions that track partitions by the number of odd columns.

If this is right

  • Non-standard formulas exist for the number of standard Young tableaux with a bounded number of columns and a fixed number of odd-length columns.
  • Identities hold between the numbers of standard Young tableaux and the numbers of marked vacillating tableaux for certain restricted shapes.
  • Combinatorial interpretations of the identities can be given in terms of up-down tableaux.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • These refinements might extend to other bounded cases or to q-analogues by modifying the skewing process accordingly.
  • Connections to representation theory could arise since Schur functions and Littlewood identities often relate to characters of symmetric groups or orthogonal groups.
  • The combinatorial models with vacillating tableaux may lead to new bijections or generating function proofs for tableau enumeration problems.

Load-bearing premise

The generating function counting partitions with a bounded number of columns and a prescribed number of odd columns can be obtained by applying skewing operators to the classical bounded Littlewood identity.

What would settle it

Direct computation of the sum of Schur functions over all partitions with at most n columns and exactly k odd-length columns for small n and k, and checking whether it equals the corresponding determinant expression for the product.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper refines bounded Littlewood identities by prescribing the number of odd-length columns (rather than rows) in partitions with a bounded number of columns, extending Goulden's 1992 result. It introduces new formulations of these identities that employ skewing operators applied to the classical bounded Littlewood product, yielding determinant representations. Corollaries include non-standard formulas for the number of standard Young tableaux of restricted shapes and combinatorial interpretations in terms of up-down tableaux and (marked) vacillating tableaux.

Significance. If the claimed identities and their derivations hold, the work supplies new algebraic tools for enumerating Young tableaux under column bounds and parity constraints, together with explicit links to vacillating tableaux. The skewing-operator reformulations may prove useful for further refinements or generalizations in the theory of symmetric functions and tableau combinatorics.

major comments (1)
  1. [§4] §4 (skewing-operator formulation): the transition from the row-parity determinant (Goulden) to the column-parity case via skewing operators is load-bearing for the main refinement. The manuscript must explicitly verify that the skewing commutes with the column bound without introducing extra terms or altering the product side, because conjugation interchanges rows and columns and the bounding condition is asymmetric. A concrete check (e.g., expansion of the first few terms or an explicit low-rank example) is required to confirm the determinant form is preserved.
minor comments (2)
  1. [Abstract / Introduction] The abstract and introduction should clarify the precise relationship between the new column-parity identities and Goulden's row-parity identities, including whether the former are obtained by conjugation or by a genuinely independent construction.
  2. [Preliminaries] Notation for the skewing operators and the precise statement of the bounded Littlewood product should be collected in a preliminary section to improve readability of the later determinant identities.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback on the manuscript. We address the single major comment below and will incorporate the requested verification in the revised version.

read point-by-point responses

  1. Referee: [§4] §4 (skewing-operator formulation): the transition from the row-parity determinant (Goulden) to the column-parity case via skewing operators is load-bearing for the main refinement. The manuscript must explicitly verify that the skewing commutes with the column bound without introducing extra terms or altering the product side, because conjugation interchanges rows and columns and the bounding condition is asymmetric. A concrete check (e.g., expansion of the first few terms or an explicit low-rank example) is required to confirm the determinant form is preserved.

    Authors: We agree that an explicit verification of the compatibility between the skewing operator and the column bound strengthens the argument, given the asymmetry under conjugation. In the revised manuscript we will insert a concrete low-rank check: we expand both sides of the relevant identity for partitions with at most two columns, up to total degree 5, and verify term-by-term that the skewing operator applied to the bounded product yields precisely the expected determinant without extraneous summands. This calculation will be placed immediately after the statement of the skewing-operator identity in Section 4. revision: yes

Circularity Check

0 steps flagged

No circularity: external citation and operator-based extension are independent of target counts

full rationale

The paper explicitly treats Goulden's 1992 result on bounded columns with prescribed odd rows as an external reference point, then applies skewing operators to obtain the odd-column refinement and determinant form. No quoted step defines the odd-column generating function in terms of the final identity, fits parameters to the target counts, or reduces the claimed determinant to a renaming of the input product. The derivation chain remains self-contained against the classical bounded Littlewood identities and the cited external work.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The identities rest on the standard ring structure of symmetric functions, the known determinant form of bounded Littlewood identities, and the definition of skewing operators; no new free parameters or invented entities are introduced in the abstract.

axioms (2)
  • standard math Schur functions form a basis of the ring of symmetric functions and satisfy the usual Cauchy identity and skewing rules.
    Invoked implicitly when the paper replaces the infinite product by a determinant and applies skewing operators.
  • domain assumption The classical bounded Littlewood identity holds when the number of rows or columns is fixed.
    Taken as the starting point for the parity refinements.

pith-pipeline@v0.9.0 · 5750 in / 1309 out tokens · 24698 ms · 2026-05-18T08:10:32.627726+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

24 extracted references · 24 canonical work pages

  1. [1]

    E. A. Bender and D. E. Knuth. Enumeration of plane partitions.J. Combin. Theory Ser. A13(1972), 40–54

    work page 1972

  2. [2]

    Eu, T.-S

    S.-P. Eu, T.-S. Fu, J. T. Hou, and T.-W. Hsu. Standard Young tableaux and colored Motzkin paths.J. Combin. Theory Ser. A 120(7) (2013), 1786–1803

    work page 2013

  3. [3]

    I. M. Gessel. Symmetric functions and P-recursiveness.J. Combin. Theory Ser. A53(2) (1990), 257–285

    work page 1990

  4. [4]

    I. M. Gessel and X. Viennot. Binomial determinants, paths, and hook length formulae,Adv. Math.58(1985), 300–321

    work page 1985

  5. [5]

    I. M. Gessel and X. Viennot. Determinants, paths, and plane partitions, preprint, 1989; available at http://www.cs.brandeis.edu/˜ira

    work page 1989

  6. [6]

    B. Gordon. Notes on plane partitions. V.J. Combin. Theory Ser. B11(1971), 157–168

    work page 1971

  7. [7]

    Gordon and L

    B. Gordon and L. Houten. Notes on plane partitions. II.J. Combin. Theory4(1968), 81–99

    work page 1968

  8. [8]

    I. P. Goulden. A linear operator for symmetric functions and tableaux in a strip with given trace.Discrete Math.99(1–3) (1992), 69–77

    work page 1992

  9. [9]

    J. Huh, J. S. Kim, C. Krattenthaler, and S. Okada. Bounded Littlewood identities for cylindric Schur functions.Trans. Amer. Math. Soc.378(2025), 6765–6829

    work page 2025

  10. [10]

    Ishikawa and M

    M. Ishikawa and M. Wakayama. Minor summation formula for pfaffians.Linear and Multilinear Algebra39(1995), 285–305

    work page 1995

  11. [11]

    J. S. Kim, K.-H. Lee, and S.-j. Oh. Weight multiplicities and Young tableaux through affine crystals.Mem. Amer. Math. Soc., vol. 283, no. 1401, 2023, Amer. Math. Soc., Providence, R.I., v+88 pp

    work page 2023

  12. [12]

    K. Koike. Representations of spinor groups and the difference characters ofS O(2n),Adv. Math.128(1997), 40–81

    work page 1997

  13. [13]

    Krattenthaler

    C. Krattenthaler. Non-crossing two-rowed arrays and summations for Schur functions, A. Barlotti, M. Delest, R. Pinzani, Proc. of the 5th Conference on Formal Power Series and Algebraic Combinatorics, Florence, 1993, D.S.I., Universit `a di Firenze, 1993, 301–314

    work page 1993

  14. [14]

    Krattenthaler, Oscillating tableaux and nonintersecting lattice paths,J

    C. Krattenthaler, Oscillating tableaux and nonintersecting lattice paths,J. Statist. Plann. Inference54(1996), 75–85

    work page 1996

  15. [15]

    Krattenthaler

    C. Krattenthaler. Identities for classical group characters of nearly rectangular shape,J. Algebra209(1998), 1–64

    work page 1998

  16. [16]

    Krattenthaler

    C. Krattenthaler. Growth diagrams, and increasing and decreasing chains in fillings of Ferrers shapes,Adv. Appl. Math.37 (2006), 404–431

    work page 2006

  17. [17]

    Krattenthaler

    C. Krattenthaler. Bijections between oscillating tableaux and (semi)standard tableaux via growth diagrams,J. Combin. Theory Ser. A144(2016), 277–291

    work page 2016

  18. [18]

    Lindstr ¨om

    B. Lindstr ¨om. On the vector representations of induced matroids,Bull. London Math. Soc.5(1973), 85–90

    work page 1973

  19. [19]

    I. G. Macdonald.Symmetric Functions and Hall Polynomials. Oxford Mathematical Monographs. The Clarendon Press Ox- ford University Press, New York, second edition, 1995

    work page 1995

  20. [20]

    S. Okada. Applications of minor summation formulas to rectangular-shaped representations of classical groups,J. Algebra 205(1998), 337–367

    work page 1998

  21. [21]

    Okada, Pieri rules for classical groups and equinumeration between generalized oscillating tableaux and semistandard tableaux,Electron

    S. Okada, Pieri rules for classical groups and equinumeration between generalized oscillating tableaux and semistandard tableaux,Electron. J. Combin.23(2016), Art. #P4.43, 27 pp

    work page 2016

  22. [22]

    R. A. Proctor. Young tableaux. Gelfand patterns and branching rules for classical Lie groups,J. Algebra164(1994), 299–360

    work page 1994

  23. [23]

    R. P. Stanley,Enumerative Combinatorics, vol. 2, Cambridge University Press, Cambridge, 1999

    work page 1999

  24. [24]

    Zeilberger

    D. Zeilberger. The number of ways of walking inx 1 ≥ · · · ≥x k ≥0 forndays, starting and ending at the origin, where at each day you may either stay in place or move one unit in any direction, equals the number ofn-cell standard Young tableaux with ≤2k+1 rows.The Personal Journal of Shalosh B. Ekhad and Doron Zeilberger, 2007. BOUNDED LITTLEWOOD IDENTITI...

    work page 2007