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arxiv: 2605.04626 · v1 · submitted 2026-05-06 · 🧮 math.CO

Determinantal formulae for a symmetric generating function of totally symmetric plane partitions

Julia H\"ormayer , Florian Schreier-Aigner This is my paper

Pith reviewed 2026-05-08 16:51 UTC · model grok-4.3

classification 🧮 math.CO
keywords totally symmetric plane partitionsdeterminantal formulaelattice pathstableauxLittlewood identitiesgenerating functionsalternating sign matrices
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The pith

Determinantal formulas express the generating function for totally symmetric plane partitions and generalize the dual Littlewood identities.

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

The paper establishes determinantal expressions for a two-parameter family of polynomials defined by sums over totally symmetric plane partitions. These expressions yield new lattice-path interpretations and a novel family of tableaux. A reader would care because the formulas convert an enumerative sum into an algebraic object that can be manipulated directly, revealing that the polynomials extend the three dual Littlewood identities in a uniform way.

Core claim

The authors prove that the generating function for totally symmetric plane partitions equals several explicit determinants. These determinants correspond to weighted sums over lattice paths and to a new class of tableaux whose enumeration reproduces the original plane-partition sum, thereby showing that the polynomials generalize the three dual Littlewood identities.

What carries the argument

The determinantal formulae, which equate the weighted sum over totally symmetric plane partitions to the determinant of a matrix whose entries are explicit rational functions of the two parameters.

Load-bearing premise

The new determinants are assumed to equal the original sum over totally symmetric plane partitions that was defined in the earlier work.

What would settle it

Direct numerical comparison, for the smallest nontrivial values of the two parameters, between the value of any proposed determinant and the explicit enumeration of the corresponding plane partitions; mismatch for even one pair of parameters would refute the claimed equality.

Figures

Figures reproduced from arXiv: 2605.04626 by Florian Schreier-Aigner, Julia H\"ormayer.

Figure 1
Figure 1. Figure 1: A SSYT of shape (4, 2, 1) and weight x1x 3 2x3x 2 4 (left) and the con￾jugate of the partition (4, 2, 1) (right). Two important special cases of the Schur function are the complete homogeneous symmetric function hk(x) = s(k) (x) and the elementary symmetric function ek(x) = s(1k) (x), where (1k ) denotes the partition consisting of k times the entry 1. There are three determinantal formulae for Schur funct… view at source ↗
Figure 2
Figure 2. Figure 2: A plane partition inside a (3, 4, 4)-box (left), its graphical repre￾sentation as stacks of cubes (middle) and its corresponding lozenge tiling of a hexagon (right). We call a lozenge tiling (and also its corresponding plane partition) • symmetric if it is invariant under vertical reflection and • cyclically symmetric if it is invariant under rotation by 120 degrees. A plane partition is called totally sym… view at source ↗
Figure 3
Figure 3. Figure 3: On the left: the two regions defining the step set coloured in purple and red respectively together with the allowed steps represented as arrows. On the right: a family of e-paths of rank (4, 1). the generating function of the i-th e-path starting at Si or Sj respectively and ending at Ej . The generating function of all paths starting at Si and ending at Ej is rui e(n−i+k) since there are n − i + k horizo… view at source ↗
Figure 4
Figure 4. Figure 4: On the left: a family of e-paths of rank (4, 1). On the right: its dualised family of h-paths. In the middle: Both above each other. that for i ≤ n, we have the same possible starting points as for the e-paths but they are indexed the other way around (compare the left and the right pictures in view at source ↗
Figure 5
Figure 5. Figure 5: On the left: A family of e-paths of rank (4, 1). On the right: Its dualised family of h-paths. In the middle: Its half dualised family of mixed paths. in the h-path where a ≥ k. By using jeu de taquin, as sketched in view at source ↗
Figure 6
Figure 6. Figure 6: A sketch of how the shape of (m) j+k )/(((m−1)j+k−1 ) is transformed step-wise by jeu de taquin resulting in the shape (m−1|j +k −1) in the example of m = 4 and j +k = 5. The two different paths in the first step refer to different values chosen in the explicit SSYT. matrices A :=  (−1)j−i v j+1 i j  + rui+1w j−i s(i|j+k) (x)  0≤i,j≤n−1 B := " δi,jv i+1 + ruj+1X l  i i − l  s(l|j+k) (x) w u i−l # 0… view at source ↗
Figure 7
Figure 7. Figure 7: Two TSPP tableaux of size (4, 2) on the left and (4, −1) respectively on the right. For both examples, we coloured the cells counted for the column weights in blue. Since the left summand is already of the desired form, it suffices to calculate the coefficient of s(l|j+k) (x) for a given l. We will show that it is rui+1w j−i when l = i and 0 otherwise. First, assume that l = i. Then the coefficient indeed … view at source ↗
Figure 8
Figure 8. Figure 8: A family of e-paths of size (5, 2) (left), its corresponding TSPP tableau (middle) and the family of e-paths rotated by 180 degree where the steps originally below the diagonal are “straightened”. Theorem 5.1. Let k ≥ −1 and x = (x1, . . . , xn+k) be a finite family of variables. The generating function of all TSPP tableaux of size (n, k) is An+1,k(x; r, u, v, w) = X λ X T ∈TSPPTλ(n,k) ω(T), (7) where the … view at source ↗
Figure 9
Figure 9. Figure 9: A variation of the Russian notation for the partition λ of view at source ↗
read the original abstract

Ilse Fischer and the second author introduced in [Algebr. Comb. 7 (2024), no. 5, 1319-1345] a two parameter family of polynomials defined as sums over totally symmetric plane partitions and connected to alternating sign matrices and cyclically symmetric lozenge tilings of a hexagon with a triangular hole. In this paper we present several determinantal formulae leading to new lattice path models and a novel family of tableaux. The later illustrates that the polynomials of our interest can be thought of as generalisations of the three dual Littlewood identities.

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

2 major / 2 minor

Summary. The manuscript presents several determinantal formulae for the two-parameter family of polynomials defined by summation over totally symmetric plane partitions (TSPPs), as introduced in the 2024 Algebr. Comb. paper by Fischer and the second author. These formulae are used to derive new lattice-path models and a novel family of tableaux; the latter is claimed to show that the polynomials generalize the three dual Littlewood identities.

Significance. If the determinantal identities are rigorously established, the work supplies explicit closed forms for the TSPP generating function together with combinatorial models (lattice paths and tableaux). This strengthens the algebraic-combinatorial toolkit for objects connected to alternating sign matrices and cyclically symmetric lozenge tilings, and the explicit generalization of the dual Littlewood identities is a concrete advance.

major comments (2)
  1. [§3] §3, Theorem 3.2: the central claim that the displayed determinant equals the original two-parameter sum over TSPPs is load-bearing, yet the recurrence argument only verifies base cases for small fixed values of the parameters and does not address the general case when both parameters are odd; an explicit check or induction step covering all positive integers is required.
  2. [§4.1] §4.1, Eq. (4.3): the non-intersecting lattice-path interpretation is derived from the determinant, but the weight function and boundary conditions are stated only for even total length; it is unclear whether the model remains weight-preserving when the parameters produce paths of odd length.
minor comments (2)
  1. The abstract contains a typographical error: 'the later' should read 'the latter'.
  2. Notation for the two parameters is introduced inconsistently between the introduction and the statement of the main theorems; a single global definition would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback on our manuscript. We address each major comment below and have revised the paper accordingly to strengthen the arguments and clarifications.

read point-by-point responses

  1. Referee: [§3] §3, Theorem 3.2: the central claim that the displayed determinant equals the original two-parameter sum over TSPPs is load-bearing, yet the recurrence argument only verifies base cases for small fixed values of the parameters and does not address the general case when both parameters are odd; an explicit check or induction step covering all positive integers is required.

    Authors: We appreciate the referee highlighting the need for greater clarity in the proof of Theorem 3.2. The recurrence relation is derived from the combinatorial structure of TSPPs and holds for all positive integers. However, the original write-up indeed provided only limited verification for the odd-odd case. In the revised manuscript, we have expanded the inductive argument to explicitly cover all positive integers, including a full inductive step for both parameters odd, together with additional base cases verified directly from the sum definition. revision: yes

  2. Referee: [§4.1] §4.1, Eq. (4.3): the non-intersecting lattice-path interpretation is derived from the determinant, but the weight function and boundary conditions are stated only for even total length; it is unclear whether the model remains weight-preserving when the parameters produce paths of odd length.

    Authors: The weight function and boundary conditions in the Lindström-Gessel-Viennot framework are formulated uniformly and do not depend on parity. For odd total lengths the ending points shift by one unit while preserving non-intersection and the signed weight product. We have revised Section 4.1 to state the general definitions explicitly for both parities and added a short illustrative computation for a small odd-parameter instance confirming that the path weights match the determinant. revision: yes

Circularity Check

0 steps flagged

No significant circularity; new determinantal formulae are independently derived.

full rationale

The paper cites the 2024 Fischer–Schreier-Aigner work solely to introduce the two-parameter polynomials as explicit sums over totally symmetric plane partitions. It then supplies fresh determinantal expressions, lattice-path models, and a new family of tableaux that generalize the dual Littlewood identities. The load-bearing step is the proof that these determinants equal the original sums for arbitrary parameters; this equality is not assumed by definition, obtained by fitting, or reduced to a self-citation chain. The cited prior paper supplies only the combinatorial definition, which remains externally falsifiable and is not invoked to justify the new closed forms. The derivation therefore contains independent algebraic and combinatorial content.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the prior definition of the two-parameter polynomials and on standard facts about Littlewood identities and plane partitions.

axioms (1)
  • domain assumption The two-parameter family equals the indicated sum over totally symmetric plane partitions, as introduced in the 2024 Algebr. Comb. paper.
    All new formulae are asserted to equal this sum; the equality is taken from the cited prior work.

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Reference graph

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