Generalized Minimal Excludants in Partition Theory

• Generalized minimal excludants are partition statistics that generalize the classic mex by modifying the definition of the first missing object using frequency thresholds, consecutive gaps, and residue restrictions.
• They yield new combinatorial identities and bijections connecting colored partitions, generalized Frobenius partitions, and q-series, enriching classical partition theory.
• Applications include analyses of parity phenomena, asymptotic moment behavior, and combinatorial proofs of truncated theta identities in various partition frameworks.

Generalized minimal excludants are partition statistics obtained by varying the classical minimal excludant, namely the smallest positive integer absent from a partition. In recent work, this variation has been carried out along several axes: absence can be replaced by multiplicity below a threshold, by a missing block of consecutive integers, by restriction to a fixed residue class, by passage to bipartitions, or by the distinction between overlined and non-overlined parts in overpartitions. The resulting statistics are tied to colored partition functions, generalized Frobenius partitions, truncated Jacobi identities, Ramanujan $q$-series, and asymptotic moment problems (Bhoria et al., 2022, Chern et al., 19 Jul 2025, Aricheta et al., 2023, Chen et al., 18 Jun 2026).

1. Classical mex and the partition-theoretic framework

For a partition $T$, the minimal excludant is

$\operatorname{mex}(T)=\min\{k\ge 1:\; k\notin T\}.$

Grabner–Knopfmacher studied this statistic under the name smallest gap, Fraenkel–Peled introduced mex in combinatorial game theory / set theory as the least missing positive integer, and Andrews–Newman adopted mex for partitions and rediscovered the smallest-gap statistic of Grabner–Knopfmacher (Kaur et al., 2021).

If $P(n)$ denotes the set of partitions of $n$, then

$$\sigma\,\mathrm{mex}(n)=\sum_{\lambda\in P(n)}\mathrm{mex}(\lambda).$$

A central identity is

$\sigma\,\mathrm{mex}(n)=D_2(n),$

where $D_2(n)$ is the number of partitions of $n$ into distinct parts using two colors. Ballantine and Merca gave a purely combinatorial proof of this result, using Ferrers diagrams, staircase partitions, and a bijection adapted from Sylvester’s bijection for Jacobi’s triple product identity. The same paper records

$$\sigma\,\mathrm{mex}(n)=\sum_{k\ge 0} p\!\left(n-\frac{k(k+1)}{2}\right),$$

and the parity statement

$T$0

for some nonnegative integer $T$1 (Ballantine et al., 2019).

This classical framework supplies the template for essentially all later generalizations: one defines a “first missing” object, forms total sums or moments over partitions of $T$2, and then seeks colored-partition interpretations, $T$3-series identities, or asymptotic formulas.

2. Frequency thresholds and missing chains

One major direction replaces “absent” by “present with frequency $T$4.” Ballantine and Merca define the least $T$5-gap of a partition $T$6 as the smallest positive integer that does not appear at least $T$7 times as a part of $T$8, and write $T$9 for the sum of these least $\operatorname{mex}(T)=\min\{k\ge 1:\; k\notin T\}.$0-gaps over all partitions of $\operatorname{mex}(T)=\min\{k\ge 1:\; k\notin T\}.$1 (Ballantine et al., 2019). In a later asymptotic formulation, the statistic is written

$\operatorname{mex}(T)=\min\{k\ge 1:\; k\notin T\}.$2

so that

$\operatorname{mex}(T)=\min\{k\ge 1:\; k\notin T\}.$3

The same paper introduces the congruence-class refinement

$\operatorname{mex}(T)=\min\{k\ge 1:\; k\notin T\}.$4

and studies moments of both $\operatorname{mex}(T)=\min\{k\ge 1:\; k\notin T\}.$5 and $\operatorname{mex}(T)=\min\{k\ge 1:\; k\notin T\}.$6 (Chern et al., 19 Jul 2025).

A second direction replaces a single missing integer by a missing block of consecutive integers. The $\operatorname{mex}(T)=\min\{k\ge 1:\; k\notin T\}.$7-chain mex of a partition $\operatorname{mex}(T)=\min\{k\ge 1:\; k\notin T\}.$8 is defined by

$\operatorname{mex}(T)=\min\{k\ge 1:\; k\notin T\}.$9

so the statistic records the first gap block of length $P(n)$0. When $P(n)$1, it is exactly the classical mex. The paper introducing this statistic proves that the number of partitions of $P(n)$2 with exactly $P(n)$3 multiples of $P(n)$4 equals the number of partitions whose largest $P(n)$5-repeating part is $P(n)$6, and also equals the number of partitions with $P(n)$7 parts greater than the $P(n)$8-chain mex. It also derives the generating function

$P(n)$9

where $n$0 is the sum of $n$1-chain mex over all partitions of $n$2 (Bhoria et al., 2022).

Bhoria, Eyyunni, and collaborators later gave combinatorial proofs of the corresponding $n$3-chain identities and introduced the complementary $n$4-chain maximal excludant. Their proofs use Glaisher’s bijection, conjugation, Ferrers-diagram cuts, and transfer operators $n$5 and $n$6, showing that the chain-based generalization fits naturally into the Euler–Glaisher–Franklin ecosystem (Bhoria et al., 2023).

3. Congruence classes and truncated theta phenomena

A third direction restricts the candidate excludants to a fixed arithmetic progression. In the notation used for truncated Jacobi identities,

$n$7

is the smallest positive integer congruent to $n$8 that is not a part of $n$9. Thus $$\sigma\,\mathrm{mex}(n)=\sum_{\lambda\in P(n)}\mathrm{mex}(\lambda).$$0 is the smallest odd positive integer absent from $$\sigma\,\mathrm{mex}(n)=\sum_{\lambda\in P(n)}\mathrm{mex}(\lambda).$$1. This generalized mex is the central combinatorial device in a partition-theoretic interpretation of bilateral truncated Jacobi triple product identities (Chen et al., 23 Jun 2026).

In that setting one considers pairs of partitions

$$\sigma\,\mathrm{mex}(n)=\sum_{\lambda\in P(n)}\mathrm{mex}(\lambda).$$2

together with refined counts determined by congruence conditions on $$\sigma\,\mathrm{mex}(n)=\sum_{\lambda\in P(n)}\mathrm{mex}(\lambda).$$3 and $$\sigma\,\mathrm{mex}(n)=\sum_{\lambda\in P(n)}\mathrm{mex}(\lambda).$$4. The main theorem expresses truncated alternating sums of $$\sigma\,\mathrm{mex}(n)=\sum_{\lambda\in P(n)}\mathrm{mex}(\lambda).$$5 as explicit sums of two mex-refined counting functions. The paper presents this as a modular/congruence-class analogue of the Andrews–Merca truncated theorem and as a combinatorial explanation for the nonnegativity of coefficients in these truncated theta products (Chen et al., 23 Jun 2026).

The frequency-threshold version also has a residue-class refinement. For nonnegative integer $$\sigma\,\mathrm{mex}(n)=\sum_{\lambda\in P(n)}\mathrm{mex}(\lambda).$$6,

$$\sigma\,\mathrm{mex}(n)=\sum_{\lambda\in P(n)}\mathrm{mex}(\lambda).$$7

and

$$\sigma\,\mathrm{mex}(n)=\sum_{\lambda\in P(n)}\mathrm{mex}(\lambda).$$8

Their generating functions are given explicitly, and the asymptotic theory shows that for fixed modulus $$\sigma\,\mathrm{mex}(n)=\sum_{\lambda\in P(n)}\mathrm{mex}(\lambda).$$9, the choice of residue class $\sigma\,\mathrm{mex}(n)=D_2(n),$0 does not matter in the leading term: $\sigma\,\mathrm{mex}(n)=D_2(n),$1 This is the precise sense in which the congruence-class moments are asymptotically “equal” (Chern et al., 19 Jul 2025).

4. Colored partitions, distinct parts, and bipartitions

Generalized minimal excludants repeatedly admit colored-partition interpretations. In the classical case, Andrews–Newman’s identity relates $\sigma\,\mathrm{mex}(n)=D_2(n),$2 to 2-colored distinct partitions, and a later refinement splits the total according to the parity of the mex. If $\sigma\,\mathrm{mex}(n)=D_2(n),$3 and $\sigma\,\mathrm{mex}(n)=D_2(n),$4 denote 2-colored distinct partitions of $\sigma\,\mathrm{mex}(n)=D_2(n),$5 with an odd or even number of parts, then

$\sigma\,\mathrm{mex}(n)=D_2(n),$6

and

$\sigma\,\mathrm{mex}(n)=D_2(n),$7

The same paper develops $\sigma\,\mathrm{mex}(n)=D_2(n),$8-th moments of mex and signed moment refinements in terms of triangular-number transforms of $\sigma\,\mathrm{mex}(n)=D_2(n),$9 (Bhoria et al., 2021).

For partitions into distinct parts, the sum

$D_2(n)$0

has generating function

$D_2(n)$1

where $D_2(n)$2 is Ramanujan’s function. The same paper also studies odd minimal excludant and maximal excludant analogues over distinct partitions, and identifies Uncu’s series with the generating function counting distinct partitions whose mex is odd (Kaur et al., 2021).

A more recent extension passes from partitions to bipartitions. For a bipartition $D_2(n)$3, Lin and Liu define

$D_2(n)$4

the smallest positive integer that does not appear simultaneously in both components. Writing

$D_2(n)$5

and

$D_2(n)$6

the 2026 paper on two-colored generalized Frobenius partitions proves

$D_2(n)$7

The proof is entirely combinatorial and proceeds by bijections built from staircase partitions and staircase insertion maps (Chen et al., 18 Jun 2026).

Generalized mex statistics also appear in two-color partitions in which even parts may occur only in blue. In that setting

$D_2(n)$8

is the smallest positive integer congruent to $D_2(n)$9 that does not occur in $n$0, and Andrews–El Bachraoui’s identities

$n$1

receive explicit combinatorial proofs via bijections built from the ordinary odd minimal excludant $n$2 and the generalized blue-part mex (Dandan et al., 27 Jun 2026).

5. Overpartition generalizations

Overpartitions support several inequivalent mex notions. Aricheta and Donato define the minimal excludant of an overpartition $n$3, denoted $n$4, to be the smallest positive integer that is not a part of the non-overlined parts of $n$5. The corresponding sum

$n$6

satisfies

$n$7

where $n$8 counts partitions of $n$9 into distinct parts using three colors. The same paper generalizes to least $$\sigma\,\mathrm{mex}(n)=\sum_{k\ge 0} p\!\left(n-\frac{k(k+1)}{2}\right),$$0-gaps $$\sigma\,\mathrm{mex}(n)=\sum_{k\ge 0} p\!\left(n-\frac{k(k+1)}{2}\right),$$1, defined as the smallest part of the non-overlined parts of $$\sigma\,\mathrm{mex}(n)=\sum_{k\ge 0} p\!\left(n-\frac{k(k+1)}{2}\right),$$2 appearing fewer than $$\sigma\,\mathrm{mex}(n)=\sum_{k\ge 0} p\!\left(n-\frac{k(k+1)}{2}\right),$$3 times, and derives a generating function for $$\sigma\,\mathrm{mex}(n)=\sum_{k\ge 0} p\!\left(n-\frac{k(k+1)}{2}\right),$$4 (Aricheta et al., 2023).

A different viewpoint treats an integer as present in an overpartition if it appears in either overlined or non-overlined form. This yields four new mex-type statistics: $$\sigma\,\mathrm{mex}(n)=\sum_{k\ge 0} p\!\left(n-\frac{k(k+1)}{2}\right),$$5, $$\sigma\,\mathrm{mex}(n)=\sum_{k\ge 0} p\!\left(n-\frac{k(k+1)}{2}\right),$$6, $$\sigma\,\mathrm{mex}(n)=\sum_{k\ge 0} p\!\left(n-\frac{k(k+1)}{2}\right),$$7, and $$\sigma\,\mathrm{mex}(n)=\sum_{k\ge 0} p\!\left(n-\frac{k(k+1)}{2}\right),$$8. The ordered variants use the total order

$$\sigma\,\mathrm{mex}(n)=\sum_{k\ge 0} p\!\left(n-\frac{k(k+1)}{2}\right),$$9

so that $T$00 is the smallest overlined positive integer that is not a part of $T$01. Their generating functions are expressed through Ramanujan’s $T$02, $T$03, the fifth-order mock theta functions $T$04 and $T$05, and corresponding arithmetic sums are realized as sum-of-tails and $T$06-harmonic constructions (Dhar et al., 2024).

Another 2025 paper isolates two additional overpartition mex notions: the mex of the overlined parts and the mex of both overlined and non-overlined parts. If $T$07 denotes the sum of the mex of the overlined parts over all overpartitions of $T$08, then

$T$09

If $T$10 denotes the analogous sum for the mex of all parts, then

$T$11

The first is governed by Ramanujan’s $T$12; the second has a basic hypergeometric representation (Donato, 6 Jul 2025).

The chain paradigm also extends to overpartitions. The $T$13-chain minimal excludant size of an overpartition $T$14,

$T$15

is the smallest positive integer $T$16 such that there are no parts of size $T$17 in $T$18. The key structural theorem states that, under conjugation of overpartitions, $T$19 is controlled by the largest $T$20-repeating size. This yields explicit generating functions for

$T$21

and for the complementary maximal statistic $T$22 (Chen et al., 22 Jan 2025).

6. Asymptotics, parity, and arithmetic distribution

Generalized minimal excludants have a substantial asymptotic theory. For the frequency-threshold and residue-class statistics, the moment generating functions are analyzed through asymptotics of weighted partial theta functions and Ingham’s Tauberian theorem. The resulting first-order asymptotics show that the leading term depends on $T$23, $T$24, and $T$25, but not on the residue class $T$26; consequently the moments are asymptotically equidistributed across congruence classes for fixed modulus (Chern et al., 19 Jul 2025).

Parity phenomena are equally prominent. For the classical sum $T$27, odd values occur exactly when

$T$28

for some nonnegative integer $T$29 (Ballantine et al., 2019). For the non-overlined overpartition mex,

$T$30

for some $T$31, so $T$32 is odd exactly at triangular numbers and therefore even for almost all $T$33. The same paper proves that if

$T$34

then for any fixed $T$35,

$T$36

(Aricheta et al., 2023).

For other overpartition mex variants, the parity behavior changes. The sum of the mex of the overlined parts is almost always even, whereas the sum of the mex of both overlined and non-overlined parts is always even for every positive integer $T$37 (Donato, 6 Jul 2025). On restricted partition classes, asymptotic formulas also remain available: for instance, the sum of mex over partitions into distinct parts is treated by a combination of bivariate generating functions, Ramanujan’s function $T$38, and Ingham’s Tauberian theorem (Kaur et al., 2021).

Taken together, these results show that generalized minimal excludants form a coherent but highly nonuniform family. The unifying principle is the replacement of the first missing positive integer by a more structured first missing object; the resulting theories differ in whether the missing object is a single part, a gap block, a frequency defect, a residue-class representative, a bipartition obstruction, or an overpartition component. This suggests a common partition-theoretic program in which excludant statistics are organized by the ambient combinatorial category and by the notion of absence imposed on that category.