Super-Partitions in Combinatorics & Supersymmetry

• Super-partitions are generalized partitions that incorporate additional parity, fermionic, and spacing data to encode supersymmetric combinatorial structures.
• They appear in diverse contexts, from quantum toroidal superalgebras and super Macdonald theory to Rogers–Ramanujan identities and superspace duality modules.
• Multiple definitions—such as parity-controlled partitions, half-integer Young diagrams, and gap-constrained partitions—link combinatorics with advanced representation theory and geometry.

Super-partitions are combinatorial generalizations of ordinary partitions in which additional parity, fermionic, superspace, or spacing data is built into the indexing object. The term is not uniform across the literature. In recent arXiv work it denotes, among other things, parity-constrained partitions and plane partitions attached to quantum toroidal superalgebras (Bezerra et al., 2021), half-integer or marked Young-diagram data indexing super Macdonald polynomials and torus fixed points on the blow-up of $\mathbb{P}^2$ (Galakhov et al., 2024, Kanno et al., 2 Jun 2025), partitions with super-distinct parts in Rogers–Ramanujan theory (Ballantine et al., 2023), and ordered set partitions with barred non-minimal elements in superspace duality modules (Rhoades et al., 2021). A related but looser usage appears in multiplicative number theory, where ordinary partitions are transported to prime factorizations by the supernorm map (Dawsey et al., 2021).

1. Terminological scope

The principal modern usages of the term are structurally different. In each case, however, the ordinary partition is modified so that an additional $\mathbb{Z}_2$-type datum affects admissibility, statistics, or representation-theoretic meaning (Bezerra et al., 2021, Galakhov et al., 2024, Ballantine et al., 2023, Rhoades et al., 2021, Dawsey et al., 2021, Kanno et al., 2 Jun 2025).

Usage	Defining feature	Main setting
$\mathbf{s}$-partition / plane $\mathbf{s}$-partition	Repetitions and stacking depend on a parity sequence $\mathbf{s}$	Quantum toroidal superalgebras
Superpartition $\Lambda$	Half-integer parts or marked removable boxes	Super Macdonald theory, blow-up geometry
Super-distinct partition	Adjacent parts differ by at least $2$	Rogers–Ramanujan identities
Ordered set superpartition	Ordered set partition with barred non-minimal elements	Superspace duality modules
Supernormal correspondence	Ordinary partitions viewed through a prime-index map	Multiplicative number theory

A common source of confusion is the assumption that all super-partitions are half-integer Young diagrams. The literature summarized here shows that no single universal definition is in force. The shared principle is instead that the indexing object is modified so that bosonic and fermionic behavior, or an additive–multiplicative correspondence, becomes combinatorially visible.

2. $\mathbf{s}$-partitions and plane $\mathbf{s}$-partitions

In the representation theory of the quantum toroidal superalgebra $\mathcal{E}_{\mathbf{s}}$ associated with $\mathbb{Z}_2$0, the parity sequence

$\mathbb{Z}_2$1

controls which repetitions in a partition-like object are admissible (Bezerra et al., 2021). For $\mathbb{Z}_2$2, a weakly decreasing sequence $\mathbb{Z}_2$3 of positive integers is an $\mathbb{Z}_2$4-partition if repetition $\mathbb{Z}_2$5 is allowed iff

$\mathbb{Z}_2$6

This produces a super-analogue of an ordinary partition in which equality of adjacent parts is parity-sensitive.

The paper interprets these objects through colored Young diagrams. Boxes carry colors in $\mathbb{Z}_2$7 according to

$\mathbb{Z}_2$8

and the wedge rule “$\mathbb{Z}_2$9 allowed iff $\mathbf{s}$0 is odd” becomes a restriction on which row endings and column endings may repeat. In the non-super case $\mathbf{s}$1, these constraints disappear and one recovers ordinary partitions.

Fock modules are indexed not by a single object but by pairs $\mathbf{s}$2 of super-partitions. For the basic pure Fock module $\mathbf{s}$3, $\mathbf{s}$4 is an $\mathbf{s}$5-partition, $\mathbf{s}$6 is an $\mathbf{s}$7-partition, and $\mathbf{s}$8. The generators $\mathbf{s}$9 and $\mathbf{s}$0 act by removing or adding colored convex and concave boxes, and the coefficients are given by Pieri-type formulas with super-signs.

The three-dimensional analogue is the plane $\mathbf{s}$1-partition, which labels bases of MacMahon modules. A basis vector is an infinite sequence of pairs

$\mathbf{s}$2

with only finitely many non-empty layers and with interlacing condition

$\mathbf{s}$3

Each horizontal slice is an $\mathbf{s}$4-partition, and a vertical parity rule governs when repetition is allowed along vertical facets. The paper summarizes this by stating that if a color can be repeated in a horizontal layer, then it cannot be repeated on the corresponding vertical facet. This is the three-dimensional super-analogue of the wedge constraint.

These combinatorial models carry explicit character formulas. For pure MacMahon modules, the character factorizes as

$\mathbf{s}$5

with $\mathbf{s}$6 encoding root parity. In the bosonic limit $\mathbf{s}$7 and $\mathbf{s}$8, $\mathbf{s}$9-partitions become ordinary partitions, plane $\mathbf{s}$0-partitions become classical plane partitions, and the character reduces to a MacMahon-type product (Bezerra et al., 2021).

3. Half-integer superpartitions and super Young diagrams

A second major usage defines a superpartition as a half-integer partition. In one formulation, a superpartition is

$\mathbf{s}$1

with integers interpreted as bosonic entries and half-integers as fermionic ones (Kanno et al., 2 Jun 2025). The strengthened ordering condition is that if $\mathbf{s}$2 is odd, then

$\mathbf{s}$3

A related formulation requires strict inequality when neighboring parts are both half-integers (Galakhov et al., 2024). In both cases the level is

$\mathbf{s}$4

Geometrically, these objects are drawn as super Young diagrams. Integer parts end with full boxes, while half-integer parts end with half-boxes or upper triangles. The generating function for superpartitions by numbers of full boxes and upper triangles is

$\mathbf{s}$5

and hence

$\mathbf{s}$6

This equals the character of a free boson Fock space times a free NS fermion Fock space (Kanno et al., 2 Jun 2025).

Two encodings are especially important. The first writes

$\mathbf{s}$7

with $\mathbf{s}$8 an ordinary partition and $\mathbf{s}$9, so that $\Lambda$0 marks a fermionic row. The second represents a superpartition by a pair $\Lambda$1, where $\Lambda$2 is an ordinary Young diagram and $\Lambda$3 is a set of removable boxes marked by $\Lambda$4. Marking a removable box subtracts $\Lambda$5 from the corresponding row length. The fermion number is $\Lambda$6, and the notations

$\Lambda$7

separate the underlying bosonic diagram from the marked-box subtraction (Kanno et al., 2 Jun 2025).

This combinatorics feeds directly into super-symmetric functions. Super-partitions index homogeneous polynomials in commuting power sums $\Lambda$8 and Grassmann variables $\Lambda$9, with $2$0 and $2$1. In this framework, ordinary partitions appear as the purely bosonic subfamily with no half-boxes (Galakhov et al., 2024).

4. Super Macdonald theory, blow-up geometry, and quantum toroidal $2$2

Super-Schur and super-Macdonald polynomials are indexed by the half-integer superpartitions just described. One construction defines super-Schur polynomials as joint eigenfunctions of commuting Cartan operators from the affine super-Yangian $2$3, and defines super-Macdonald polynomials $2$4 by row-triangularity in the super-Schur basis together with orthogonality for a $2$5-deformed scalar product (Galakhov et al., 2024). A notable structural feature is that the same coefficients also give a column-triangular expansion, extending the classical Macdonald two-order phenomenon to the super setting.

The blow-up geometry of $2$6 provides a second realization. Fixed points of the torus action on the moduli space $2$7 of framed stable perverse coherent sheaves are in bijection with $2$8-tuples $2$9, hence with $\mathbf{s}$0-tuples of superpartitions (Kanno et al., 2 Jun 2025). The total number of marked boxes satisfies

$\mathbf{s}$1

so the markings encode the exceptional-curve contribution.

For a single superpartition $\mathbf{s}$2, the paper distinguishes relevant and irrelevant boxes. Irrelevant boxes consist of the marked boxes together with one additional box for each unordered pair of marked boxes, giving

$\mathbf{s}$3

The complement $\mathbf{s}$4 of relevant boxes controls both the super Macdonald norm and the geometric fixed-point formulas.

The equivariant tangent character at a fixed point determines a Nekrasov factor $\mathbf{s}$5 for a pair of superpartitions. When $\mathbf{s}$6, this reduces to the ordinary Nekrasov factor for ordinary partitions. In the super case there is an additional fermionic correction from the marked boxes. These factors compute localization contributions to the partition function on the blow-up and also matrix elements of the quantum toroidal algebra of type $\mathbf{s}$7 on the $\mathbf{s}$8-group of the moduli space (Kanno et al., 2 Jun 2025).

The Pieri rules make the super combinatorics explicit. The zero modes $\mathbf{s}$9 and $\mathbf{s}$0 act by adding a fermionic tile to a bosonic row or by replacing a fermionic marking by an ordinary box. After normalization, the coefficients are

$\mathbf{s}$1

and

$\mathbf{s}$2

A central result is that these coefficients are recovered from residues of ratios of Nekrasov factors, so the sign $\mathbf{s}$3 acquires a direct geometric interpretation (Kanno et al., 2 Jun 2025).

5. Super-distinct partitions in Rogers–Ramanujan theory

In another established usage, a super-partition is a partition with super-distinct parts, meaning that any two successive parts differ by at least $\mathbf{s}$4: $\mathbf{s}$5 This is the same as the “2-distinct” condition in the abstract (Ballantine et al., 2023). Here the adjective “super” refers not to parity labels or Grassmann variables but to a spacing constraint.

These partitions are one side of the Rogers–Ramanujan identities. The first identity equates partitions of $\mathbf{s}$6 into parts congruent to $\mathbf{s}$7 with partitions of $\mathbf{s}$8 having super-distinct parts, while the second equates partitions into parts congruent to $\mathbf{s}$9 with partitions having super-distinct parts greater than $\mathcal{E}_{\mathbf{s}}$0 (Ballantine et al., 2023). The standard generating series are

$\mathcal{E}_{\mathbf{s}}$1

for super-distinct partitions, and

$\mathcal{E}_{\mathbf{s}}$2

for super-distinct partitions with smallest part $\mathcal{E}_{\mathbf{s}}$3.

A useful structural decomposition is that a partition with $\mathcal{E}_{\mathbf{s}}$4 super-distinct parts can be written uniquely as

$\mathcal{E}_{\mathbf{s}}$5

with $\mathcal{E}_{\mathbf{s}}$6 a partition having at most $\mathcal{E}_{\mathbf{s}}$7 parts. The odd staircase $\mathcal{E}_{\mathbf{s}}$8 is exactly what forces the gap-$\mathcal{E}_{\mathbf{s}}$9 condition.

The paper studies not only counting but also the total number of parts across these families. For the first Rogers–Ramanujan pair, the excess of the total number of parts in partitions with parts $\mathbb{Z}_2$00 over the total number of parts in super-distinct partitions of $\mathbb{Z}_2$01 equals the number of pairs $\mathbb{Z}_2$02 such that $\mathbb{Z}_2$03 is super-distinct, $\mathbb{Z}_2$04, $\mathbb{Z}_2$05, and if $\mathbb{Z}_2$06 then at least one of $\mathbb{Z}_2$07, $\mathbb{Z}_2$08, or $\mathbb{Z}_2$09 is a part of $\mathbb{Z}_2$10 (Ballantine et al., 2023). For the second Rogers–Ramanujan pair, an analogous excess is described as the cardinality of a complement $\mathbb{Z}_2$11, where the sets $\mathbb{Z}_2$12 are defined by a case-by-case injection from marked super-distinct partitions greater than $\mathbb{Z}_2$13.

This line of work places super-distinct partitions in the same “companion identity” tradition as the Andrews–Beck refinements of Euler’s identity, but with the Rogers–Ramanujan congruence classes replacing the odd/distinct dichotomy.

6. Ordered set superpartitions and superspace duality modules

A different combinatorial family appears in the superspace ring

$\mathbb{Z}_2$14

where the $\mathbb{Z}_2$15 commute and the $\mathbb{Z}_2$16 anticommute (Rhoades et al., 2021). Here a set superpartition of $\mathbb{Z}_2$17 is a set partition $\mathbb{Z}_2$18 in which non-minimal elements may be barred or unbarred, but the minimal element of each block must be unbarred. An ordered set superpartition is an ordered tuple

$\mathbb{Z}_2$19

The family $\mathbb{Z}_2$20 of ordered set superpartitions of $\mathbb{Z}_2$21 into $\mathbb{Z}_2$22 blocks satisfies

$\mathbb{Z}_2$23

where $\mathbb{Z}_2$24 denotes those with exactly $\mathbb{Z}_2$25 barred elements. The bars encode the exterior-algebra contribution.

These objects index monomial bases of the doubly graded quotients $\mathbb{Z}_2$26 associated with a superspace Vandermonde $\mathbb{Z}_2$27. The construction proceeds via a coinversion code

$\mathbb{Z}_2$28

which is a bijection onto a family of substaircase words. These words correspond to monomials in $\mathbb{Z}_2$29, and the resulting substaircase monomials descend to a monomial basis of $\mathbb{Z}_2$30, hence of $\mathbb{Z}_2$31 when $\mathbb{Z}_2$32 (Rhoades et al., 2021).

The bigraded Hilbert series is therefore a generating function over ordered set superpartitions: $\mathbb{Z}_2$33 A second statistic $\mathbb{Z}_2$34 yields the same generating function. On the representation-theoretic side, the bigraded Frobenius image equals symmetric functions built from inverse descent sets of reading words of ordered set superpartitions.

The modules $\mathbb{Z}_2$35 satisfy a bigraded version of Poincaré duality. If

$\mathbb{Z}_2$36

then $\mathbb{Z}_2$37 is one-dimensional and multiplication induces perfect pairings

$\mathbb{Z}_2$38

The combinatorial statistics on ordered set superpartitions mirror this duality through an involution that complements coinversion number and inverse descent data (Rhoades et al., 2021).

7. Supernormal correspondences and comparative perspective

The paper on the supernorm introduces a different use of “super” that does not define a new decorated partition class. For a partition

$\mathbb{Z}_2$39

the supernorm is

$\mathbb{Z}_2$40

where $\mathbb{Z}_2$41 is the $\mathbb{Z}_2$42-th prime (Dawsey et al., 2021). Its inverse sends

$\mathbb{Z}_2$43

to the pre-partition

$\mathbb{Z}_2$44

This gives a monoid isomorphism between partition multiplication by multiset union and ordinary integer multiplication.

In this setting the partitions remain ordinary; the “super” aspect lies in the prime-index substitution. The resulting dictionary identifies distinct parts with squarefree integers, $\mathbb{Z}_2$45 with $\mathbb{Z}_2$46, and the partition Möbius function with the classical Möbius function. The paper explicitly notes that, in this sense, “super-partitions” are ordinary partitions viewed through the supernorm (Dawsey et al., 2021).

Taken together, these literatures show that super-partitions are best understood as a family of context-dependent supersymmetric or parity-sensitive combinatorial models rather than as a single object. In quantum toroidal representation theory, the essential feature is parity-controlled repetition and stacking. In super Macdonald theory and blow-up geometry, it is half-boxes or marked removable boxes. In Rogers–Ramanujan theory, it is gap-$\mathbb{Z}_2$47 distinctness. In superspace duality modules, it is the barring of non-minimal block elements. In supernormal correspondences, it is a transport of ordinary partitions into multiplicative number theory. The recurrence of the term therefore reflects a common thematic role—encoding extra fermionic, parity, or super-structural data—rather than a single canonical definition.