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Partitions with parity restrictions: a bijective approach

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Pith's one-line read Many parity-restricted partition identities that were proved by generating-function algebra admit direct bijective proofs, often simpler ones.

desk verdict Solid collection of explicit bijections for known parity-restricted partition identities, plus one clean modular generalization; residual left-to-reader checks are minor. read the letter →

arxiv 2607.03293 v1 pith:WIY7YO6H submitted 2026-07-03 math.CO math.NT

William Keith (Michigan Technological Universit) , Bruce E. Sagan (Michigan State University) This is my paper · ORCID
classification math.COmath.NT MSC 11P8411P8305A1705A19
keywords bijectionintegerpartitionlatticepathmockthetafunctionoverpartitionparitystandardYoungtableau
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper shows that a range of recent counting identities for integer partitions whose parts obey parity conditions can be established by explicit bijections rather than by algebraic manipulation of generating functions. The families treated include partitions with even parts all smaller (or all larger) than the odd parts, multiplicity-restricted variants of those, certain overpartitions, the coefficients of the third-order mock theta function, partitions with distinct versus unrestricted parts of each parity, triples of partitions with parity conditions, and standard Young tableaux whose shapes satisfy parity constraints arising from Motzkin and Riordan paths. Explicit maps are constructed using operations on Young diagrams, lattice paths, hooks, double rectangles, and colorings; several of the maps are simpler than the original analytic arguments. The work therefore replaces formal power-series identities with combinatorial correspondences that make the equalities visible by hand.

What carries the argument

Explicit, invertible maps built from local operations on Young diagrams, lattice-path run sequences, double rectangles, double hooks, and two-colorings that preserve the given parity and multiplicity conditions and therefore equate the relevant counting sequences.

What would settle it

For any single identity (for example Theorem 2.1 or 3.2), compute both sides by exhaustive enumeration for all n up to a few hundred and check whether the claimed bijection pairs every object on one side with a unique object on the other; a mismatch for any n falsifies the map.

Watch

Extended reading notes

Core claim

A collection of identities previously obtained by generating-function algebra for partitions (and related objects) with parity restrictions on parts or shapes all admit bijective proofs; in several cases the bijections are shorter or more transparent than the original arguments.

Load-bearing premise

That every map defined by those local diagram or path operations is well-defined (preserves the parity and multiplicity restrictions) and is inverted by the stated reverse construction, several of which are left as exercises.

Editorial extensions

If this is right

  • Equalities such as #P_o^e(n) = #P_{e,1}(n) = #P_p(n) and the parity of v_o^e(n) become visible by direct matching of diagrams rather than by series identities.
  • The same lattice-path and conjugation arguments immediately yield the corresponding statements for overpartitions.
  • The involution on partition triples extends, by cyclic rotation of p-tuples, to congruences modulo any prime.
  • The Motzkin-to-SYT and Riordan-to-SYT maps give combinatorial interpretations of Catalan, Motzkin and Riordan numbers in terms of shapes with at most three rows and parity constraints on row lengths.
  • Open bijective problems listed in the final section (mock-theta class D, remaining distinct-versus-repeated inequalities, injections for n ≡ 0 mod 4) become concrete targets for further combinatorial work.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Once the maps are verified, the same diagram operations can be refined by tracking extra statistics (largest even part, number of odd parts, Durfee size) to produce multi-variable refinements of the original generating-function identities.
  • The lattice-path characterization of V_o^e suggests that other run-length conditions on Ferrers diagrams may likewise convert algebraic partition identities into conjugation or involution arguments.
  • The cyclic-group action used for prime-modulus triple congruences is a general template that could be applied to any family closed under cyclic permutation of components.
  • A successful bijection for the remaining mock-theta class D would simultaneously clarify the combinatorial meaning of the two-variable identity of Andrews–Yee and of Chern’s bipartition map.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 5 minor

Summary. The paper supplies explicit bijective proofs of a collection of known equinumerosities and congruences for partitions (and overpartitions, tableaux, triples) whose parts obey parity or multiplicity restrictions. The identities treated include Andrews’ results on P_o^e and P_e^o (Theorems 2.1–2.2), Chern’s and Passary’s statements on V_o^e (Theorems 3.2–3.4), a restricted-overpartition identity of Banerjee–Bringmann–Dixit (Theorem 4.1), three combinatorial models for the coefficients of the third-order mock theta function u(−q) (Theorem 5.2), a generating-function identity of Bringmann–Jennings-Shaffer (Theorem 6.1), a parity congruence for partition triples that generalizes Guadalupe’s result (Theorems 7.1–7.4), and several statements equating Motzkin/Riordan numbers with sums of f^λ over shapes of bounded length or parity-restricted row lengths (Theorems 8.2–8.5). Each map is defined by local, reversible operations on Young diagrams, lattice paths, hooks, double rectangles or colorings, and is accompanied by an inverse construction or an involution argument.

Significance. The work converts a series of generating-function identities that have appeared in the recent literature into transparent combinatorial statements. The lattice-path characterization of V_o^e (Lemma 3.1) and the involution on self-conjugate members that proves Passary’s parity result (Theorem 3.4) are particularly clean; the group-action argument that lifts Guadalupe’s mod-2 congruence to an arbitrary prime (Theorem 7.3) is a useful general template. The paper also isolates several open bijective problems (the remaining inequality of Bringmann–Craig–Nazaroglu, a map for the fourth model of u(−q), a direct injection for Chern’s mod-4 difference) that are now well-posed. The contribution is solid combinatorial exposition rather than a single deep new theorem, but it is of clear value to the partition-theory community.

minor comments (5)
  1. Several inverse maps and well-definedness arguments are left to the reader (Theorems 2.1, 2.2, 3.4, 4.1, 5.2, 6.1). While the omitted checks are elementary, a short sentence confirming that the inverse lands in the claimed set would improve readability.
  2. Typographical slips: “funci-tons” (p. 2), “ket” for “let” (p. 2), “corrseponding” (p. 10), “tabeau” (p. 22), “Fibure” (p. 22), “Partity” (section title 8). A light copy-edit will remove them.
  3. Figure 1 and the accompanying text refer to “the bottom line” after applying g; the figure itself shows only two rows. Clarifying the layout would help.
  4. In the statement of Theorem 2.2 the set P_e(n-1) is empty when n is even; the authors handle the cases correctly, but a parenthetical remark would prevent momentary confusion.
  5. The open problems collected in §9 are well-chosen; a one-sentence pointer to the most accessible of them (e.g., the remaining inequality of BCN25) in the introduction would strengthen the paper’s forward-looking aspect.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: all claimed identities are established by explicit, self-contained bijections on Young diagrams, lattice paths, hooks and colorings; no identity is assumed then re-derived.

full rationale

The paper’s central claim is that a collection of known equinumerosities (Andrews, Chern, Passary, Banerjee–Bringmann–Dixit, Bringmann–Jennings-Shaffer, Guadalupe, Matsakis–Vendervelde, Hemmer–Straub–Westrem) admit direct bijective proofs. Each proof constructs an explicit, reversible map (subtracting 1 from odd parts, conjugating lattice paths, adding/removing double rectangles or double hooks, recoloring, Robinson–Schensted insertion of horizontal steps, cyclic group action on tuples, etc.) and verifies that the map preserves the stated parity/multiplicity conditions and is invertible. No generating-function identity is taken as an axiom and then “proved”; the generating functions appear only as historical motivation. Self-citations (Sagan’s earlier combinatorial work) are independent of the present identities and are not load-bearing. Residual “left-to-reader” inverse checks are elementary and do not hide non-invertibility. Consequently the derivation chain never reduces to its own inputs by construction, and the circularity score is zero.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

Pure combinatorial paper. No free parameters are fitted. All background facts are standard partition theory (Young diagrams, conjugation, lattice-path encodings, Robinson–Schensted, cyclic group actions). The only “entities” introduced are the usual restricted partition sets defined for the statements; they have independent combinatorial meaning outside the paper.

assumptions (4)
  • standard math Standard facts about integer partitions, Young diagrams, conjugation, and generating functions (Andrews, The Theory of Partitions).
    Invoked throughout §§1–8 as background; never proved.
  • domain assumption Lattice-path encoding of a partition (E/N runs, coordinates, reversal) and the three elementary observations (P1)–(P3).
    Used as the combinatorial model for multiplicity restrictions in §3; taken as immediate from the embedding of the Ferrers diagram.
  • standard math Robinson–Schensted insertion and its inverse preserve the property of being a standard Young tableau.
    Used without proof in the Motzkin-path bijection of Theorem 8.2.
  • standard math Action of the cyclic group of prime order p on p-tuples by rotation has only fixed points of period 1 or p.
    Used to lift Guadalupe’s mod-2 congruence to arbitrary primes (Theorem 7.3).

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Pith. "Pith review of Partitions with parity restrictions: a bijective approach." pith.science (2026). https://pith.science/paper/WIY7YO6H

@misc{pith2026260703293,
  author       = {Pith},
  title        = {Pith review of: Partitions with parity restrictions: a bijective approach},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WIY7YO6H}},
  note         = {Machine review of arXiv:2607.03293}
}
read the original abstract

There has been recent interest in integer partitions whose parts satisfy parity restrictions: for example, those where all the odd parts are distinct, or those where all the even parts are larger than the odd parts. Often results about such partitions have been obtained by algebraic manipulation of generating functions. We show that a number of these identities can be proved in a bijective, and sometimes simpler, manner.

Figures

Figures reproduced from arXiv: 2607.03293 by the authors.

Figure 1
Figure 1. The maps f : P o e (n) → Pe,1(n) and g : Pe,1(n) → Pp(n) of Theorem 2.1 When the following statement is given in Andrews’ paper [And19], the condition of having at least one odd part is missing. Andrews gives a proof of it which combines his anti￾telescoping method [And13] with a bijection. Our proof will be purely bijective. Note that we will continue to use the notation f for our bijection in the next result, but … view at source ↗
Figure 2
Figure 2. The map f : Po(n) → Pˆe o (n) ⊎ Pe(n) ⊎ Pe(n − 1) of Theorem 2.2 If m1(λ) < ℓ(λ)/2 then, by the way the 1’s are added to the parts of λ, the partition µ will no longer have any 1’s and its smallest part will be odd. Similar reasoning shows that all even parts will be greater than all odd parts. And clearly |µ| = |λ| = n. It follows that µ ∈ Pˆe o (n). On the other hand, if m1(λ) ≥ ℓ(λ)/2 then every nonzero part of µ… view at source ↗
Figure 3
Figure 3. The lattice path P(6, 4, 3, 3) 3 Multiplicity restrictions We will now give a simple bijective proof of a result of Chern [Che21]. Let V o e (n) be the set of all λ ∈ P o e (n) satisfying the following two parity restrictions. (V1) All odd parts have even multiplicity. (V2) If even parts exist, then the largest has odd multiplicity and the rest even multiplicity. It will be convenient to let L(λ) = the largest even … view at source ↗
Figures from the paper (6 more)
Figure 4
Figure 4. Figure 4: The partitions ϕ 2 and ϕ˜1 The Young diagram of ϕ 2 is shown on the left in [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: Subtracting and adding a double rectangle in [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: Subtracting and adding a double hook in λ = (7, 7, 7, 7, 3, 3, 3) Theorem 3.4 ([Pas19]). For n ≥ 0 and k ≥ 1 we have v o e (n) ≡  1 (mod 2) if n = 4k(3k ± 1), 0 (mod 2) else. Proof. Consider the self-conjugate partitions in V o e (n): Vˆ o e (n) = {λ ∈ V o e (n) | λ t…
Figure 7
Figure 7. Figure 7: The map f : B(n) → A(n) of Theorem 5.2 The following result is mentioned in [And18] based on statements in the Online Encyclopedia of Integer Sequences. We note that a map similar to the one we use for proving #B(n) = #C(n) was employed by Chern [Che19b] in giving a co…
Figure 8
Figure 8. Figure 8: The map g : B(n) → C(n) of Theorem 5.2 µ = f(T λ ) in the following way. We will replace each cell of T λ other than (1, 1), which is invariant, by a set of cells which we will call a tile. Specifically, each (i, j) ∈ λ/(1, 1) is replaced by (T1) a horizontal tile of d…
Figure 9
Figure 9. Figure 9: An example of the map m : M(9) → T (9) Proof. Let T (n) = {T | T ∈ SYT(λ) where λ ⊢ n and ℓ(λ) ≤ 3}. To give a bijection m : M(n) → T (n), take P ∈ M(n) and label its steps with the elements of [n] left to right. An example of the construction of the map is given in […

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