Two-Color Partitions: Theory and Applications

• Two-color partitions are combinatorial structures where each part or point is assigned one of two colors, with different models imposing restrictions like parity, multiplicity, or difference conditions.
• Their generating functions and congruence relations connect q-series identities, Ramanujan-type arithmetic progressions, and mock theta functions, providing insight into classical and modern partition theory.
• Two-colored set partitions extend to quantum-group theory, where color operations and tensor products yield novel invariants and symmetry classifications in easy quantum groups.

Two-color partitions are combinatorial structures in which a partition carries one of two colors on each part or point. In additive number theory and $q$-series, a two-color partition of an integer $n$ is a partition in which each part is assigned a color, typically blue/red or blue/green, with parts of the same size but different colors treated as distinct and with restrictions such as “one color appears only on multiples of $k$,” “even parts occur only in blue,” or “the smallest part is odd” (Chern, 2015, Andrews et al., 2024). In a different but established usage, the term also denotes set partitions whose points are colored black or white and arranged in rows, a framework central to the theory of easy quantum groups (Mang et al., 2019, Gromada, 2018).

1. Definitions and principal models

For integer partitions, one standard family is $p_k(n)$, the number of $2$-color partitions of $n$ where one of the colors appears only in parts that are multiples of $k$. Its generating function is

$$\sum_{n \ge 0} p_k(n)q^n=\frac{1}{(q;q)_\infty (q^k;q^k)_\infty}, \qquad (a;q)_\infty=\prod_{n=0}^\infty (1-aq^n),$$

and the variant $p_{1,\ell}(n)$ is defined analogously for a general integer $\ell\ge 1$ (Chern, 2015, Gireesh et al., 2 Apr 2025). Another major model is $n$0, the set of two-color partitions in which even parts may occur only in blue; its counting function $n$1 satisfies

$n$2

Distinct-part versions, parity-refined versions, and smallest-part-restricted versions are studied extensively in recent work (Andrews et al., 29 Dec 2025, Dandan et al., 27 Jun 2026).

Difference-condition models form a separate but closely related branch. For a positive integer $n$3, the set $n$4 consists of two-color partitions of $n$5 into numerically distinct parts such that each red part is at least $n$6 larger than the next largest part, each green part is at least $n$7 larger than the next largest part, and neither $n$8 nor $n$9 is allowed as a part; the refined count $k$0 records exactly $k$1 red parts and $k$2 green parts (Fu, 2022). Other models impose Rogers–Ramanujan-type difference conditions, smallest-part restrictions, or sign weights $k$3 depending on the number of even parts (Dabbagh, 2022, Andrews et al., 2024, Andrews et al., 12 Jul 2025).

In the easy-quantum-group literature, a two-colored partition is instead a set partition built from a lower row and an upper row of colored points. Blocks may be singletons, pairs, through blocks, or non-through blocks; the category operations include tensor product, composition, involution, rotation, reflection, and related color operations (Mang et al., 2019, Mang et al., 2020). This usage is structurally different from integer partition enumeration, although both are genuinely “two-colored partition” theories.

2. Generating functions and enumerative frameworks

The generating function

$k$4

is the basic analytic object for restricted-color multiplicative models. For the case emphasized in current congruence theory,

$k$5

and the same form with $k$6 replaced by $k$7 underlies parallel results modulo powers of $k$8 (Gireesh et al., 2 Apr 2025, C. et al., 13 Mar 2025). These product forms place two-color partitions directly inside the classical $k$9-quotient and dissection machinery of partition arithmetic.

For the model with even parts blue only,

$p_k(n)$0

and the literature develops several refinements. The functions $p_k(n)$1 and $p_k(n)$2 split the count according to the parity of the number of odd red parts, while $p_k(n)$3 and $p_k(n)$4 split according to the parity of the number of even parts; their generating functions are given explicitly in product or theta-series form (Andrews et al., 29 Dec 2025, Bardhan et al., 8 Jul 2026). Distinct-part analogues produce the counting functions $p_k(n)$5, $p_k(n)$6, $p_k(n)$7, $p_k(n)$8, and $p_k(n)$9 for two-color partitions with even parts allowed only in blue (Bugleev, 27 Aug 2025).

Smallest-part-restricted families yield further generating functions with mock-theoretic content. In one line of work, $2$0 is the generating function for two-color partitions into distinct parts and is also Andrews’ generating function $2$1 for strictly concave compositions (Andrews et al., 29 Jun 2026). In another, weighted generating functions for two-color partitions with odd smallest part are identified with the third order mock theta functions $2$2, $2$3, and $2$4 (Andrews et al., 2024).

3. Arithmetic congruences and Ramanujan-type progressions

Two-color partitions support Ramanujan-type congruences closely paralleling the classical theory of $2$5. A foundational result proved that

$2$6

for $2$7, and also established

$2$8

(Chern, 2015). A subsequent classification of the modulo $2$9 progression $n$0 for $n$1 showed that the congruence holds exactly for

$n$2

and fails for all remaining $n$3 in that interval (Ghoshal et al., 2017).

The arithmetic theory has recently been extended from fixed moduli to prime-power towers. For $n$4, infinite families of Ramanujan-type congruences modulo powers of $n$5 are established in explicit arithmetic progressions, using generating-function dissections, matrix recurrences, and lower bounds on $n$6-adic valuations (C. et al., 13 Mar 2025). For $n$7, the corresponding theory yields congruences modulo $n$8, $n$9, and higher powers, again in explicit arithmetic progressions $k$0 (Gireesh et al., 2 Apr 2025).

The proof technology in the $k$1-power case combines generating function manipulations, recursive construction of coefficient vectors, Garvan’s “huffing” operator, explicit recursion formulas for matrices $k$2, and $k$3-adic valuation estimates (Gireesh et al., 2 Apr 2025). The papers on $k$4 and $k$5 are structurally parallel: both organize the generating function into progressions indexed by powers of the prime and then prove divisibility by controlling the relevant $k$6-adic orders (C. et al., 13 Mar 2025, Gireesh et al., 2 Apr 2025).

4. Overpartitions, parity refinements, and minimal excludants

A central theme in the modern theory is the equivalence between certain two-color partition counts and overpartition counts. For $k$7, the set of two-color partitions of $k$8 into distinct parts with even parts only allowed in blue, one has

$k$9

where $$\sum_{n \ge 0} p_k(n)q^n=\frac{1}{(q;q)_\infty (q^k;q^k)_\infty}, \qquad (a;q)_\infty=\prod_{n=0}^\infty (1-aq^n),$$0 counts overpartitions of $$\sum_{n \ge 0} p_k(n)q^n=\frac{1}{(q;q)_\infty (q^k;q^k)_\infty}, \qquad (a;q)_\infty=\prod_{n=0}^\infty (1-aq^n),$$1 into odd parts (Bugleev, 27 Aug 2025, Chen et al., 16 Sep 2025). Moreover,

$$\sum_{n \ge 0} p_k(n)q^n=\frac{1}{(q;q)_\infty (q^k;q^k)_\infty}, \qquad (a;q)_\infty=\prod_{n=0}^\infty (1-aq^n),$$2

with parallel formulas for $$\sum_{n \ge 0} p_k(n)q^n=\frac{1}{(q;q)_\infty (q^k;q^k)_\infty}, \qquad (a;q)_\infty=\prod_{n=0}^\infty (1-aq^n),$$3 and $$\sum_{n \ge 0} p_k(n)q^n=\frac{1}{(q;q)_\infty (q^k;q^k)_\infty}, \qquad (a;q)_\infty=\prod_{n=0}^\infty (1-aq^n),$$4 involving $$\sum_{n \ge 0} p_k(n)q^n=\frac{1}{(q;q)_\infty (q^k;q^k)_\infty}, \qquad (a;q)_\infty=\prod_{n=0}^\infty (1-aq^n),$$5 (Bugleev, 27 Aug 2025). The square exceptions are explained combinatorially by two fixed points of an almost-involution on two-modular diagrams (Bugleev, 27 Aug 2025).

For the evens-in-blue model,

$$\sum_{n \ge 0} p_k(n)q^n=\frac{1}{(q;q)_\infty (q^k;q^k)_\infty}, \qquad (a;q)_\infty=\prod_{n=0}^\infty (1-aq^n),$$6

where $$\sum_{n \ge 0} p_k(n)q^n=\frac{1}{(q;q)_\infty (q^k;q^k)_\infty}, \qquad (a;q)_\infty=\prod_{n=0}^\infty (1-aq^n),$$7 denotes the overpartition function (Andrews et al., 29 Dec 2025, Dandan et al., 27 Jun 2026). In the parity split by the number of odd red parts,

$$\sum_{n \ge 0} p_k(n)q^n=\frac{1}{(q;q)_\infty (q^k;q^k)_\infty}, \qquad (a;q)_\infty=\prod_{n=0}^\infty (1-aq^n),$$8

with $$\sum_{n \ge 0} p_k(n)q^n=\frac{1}{(q;q)_\infty (q^k;q^k)_\infty}, \qquad (a;q)_\infty=\prod_{n=0}^\infty (1-aq^n),$$9 if $p_{1,\ell}(n)$0 is odd (Andrews et al., 29 Dec 2025, Bardhan et al., 8 Jul 2026). The same line of work proves congruence criteria for $p_{1,\ell}(n)$1 and $p_{1,\ell}(n)$2 modulo $p_{1,\ell}(n)$3, $p_{1,\ell}(n)$4, and $p_{1,\ell}(n)$5, showing in particular that modulo $p_{1,\ell}(n)$6 these functions are nonzero precisely when $p_{1,\ell}(n)$7 is a perfect square or twice a perfect square (Bardhan et al., 8 Jul 2026).

Minimal-excludant statistics provide another refinement. For partitions in $p_{1,\ell}(n)$8, the blue minimal excludant $p_{1,\ell}(n)$9 is the smallest positive integer congruent to $\ell\ge 1$0 that does not occur in blue. In this setting,

$\ell\ge 1$1

linking parity-refined two-color counts with minimal-excludant enumerants (Andrews et al., 29 Dec 2025, Dandan et al., 27 Jun 2026). Recent combinatorial proofs replace analytic $\ell\ge 1$2-series arguments by explicit weight-preserving bijections and parity-reversing involutions (Dandan et al., 27 Jun 2026).

5. Difference conditions, rank statistics, and mock theta functions

Difference-condition theories for two-color partitions were substantially developed in work on Andrews’ theorems. For $\ell\ge 1$3, $\ell\ge 1$4 equals the number of two-color partitions where parts of the same color are distinct and all green parts are even; the refinement $\ell\ge 1$5 tracks the numbers of red and green parts (Fu, 2022). For $\ell\ge 1$6, $\ell\ge 1$7 equals the number of basis partitions, and for $\ell\ge 1$8, $\ell\ge 1$9 equals the number of partitions into distinct parts, with refinements expressed in terms of Durfee-square data (Fu, 2022). The proofs use generating functions, profiles of partitions, 2-indented Ferrers diagrams, and explicit bijections.

Rogers–Ramanujan-type coloring rules yield another family. A $n$00-colored Rogers–Ramanujan partition is one in which, for each color, the difference between every two consecutive parts of the same color is at least $n$01, and no part occurs in both colors. If $n$02 denotes the number of such partitions, then

$n$03

and one has the correspondence

$n$04

with a special class of overpartitions (Dabbagh, 2022).

Two-color partitions also arise from partition rank and mock theta functions. One paper proves that the number $n$05 of partitions of $n$06 with positive odd rank equals the number $n$07 of two-color partitions in which the smallest part is even, say $n$08, and all red parts are even and lie within the interval $n$09 (Andrews et al., 13 Jan 2025). This identity leads to a new representation for the third order mock theta function $n$10, and an odd-smallest-part variant brings in another third order mock theta function $n$11 (Andrews et al., 13 Jan 2025). Complementarily, partitions with odd smallest part and parity-color restrictions give weighted generating functions equal to $n$12, $n$13, and $n$14, showing that third order mock theta functions encode explicit two-color partition classes (Andrews et al., 2024).

6. Weighted series, positivity, residue restrictions, and distinct-part companions

A recent analytic direction studies signed generating functions in which each partition contributes $n$15 according to the number of even parts. For families $n$16 and $n$17, the coefficients $n$18 and $n$19 measure the difference between even-parity and odd-parity subfamilies (Andrews et al., 12 Jul 2025). Positivity is proved for

$n$20

in the $n$21-family and for $n$22 in the $n$23-family, yielding inequalities such as $n$24 in those cases (Andrews et al., 12 Jul 2025). The same paper records that $n$25 is not positive, with first negative coefficients at $n$26 (Andrews et al., 12 Jul 2025).

Residue-theoretic support restrictions form a different refinement. For

$n$27

which arises from two-color partitions with odd smallest part, the coefficient of $n$28 vanishes for every $n$29, and more sharply for

$n$30

(Singh, 12 Jun 2026). The argument uses explicit quadratic exponent forms obtained from Bailey-transform formulas and is entirely residue-theoretic (Singh, 12 Jun 2026).

For the distinct-part series

$n$31

the coefficients $n$32 satisfy

$n$33

so nonzero coefficients modulo $n$34 occur exactly at triangular numbers (Andrews et al., 29 Jun 2026). The same work proves that

$n$35

if and only if $n$36 is not a square mod $n$37, and studies the eta-normalized odd companion $n$38, for which

$n$39

and $n$40 only if $n$41 is represented by $n$42 (Andrews et al., 29 Jun 2026).

7. Two-colored set partitions and easy quantum groups

In the quantum-group literature, two-colored partitions are set partitions of colored points rather than integer partitions. A partition consists of two ordered rows of black and white points, partitioned into blocks; blocks may be through blocks or non-through blocks, and the basic operations are monoidal product, vertical composition, involution, rotations, verticolor reflection, and erasing turns (Mang et al., 2019). Categories of such partitions are divided into four cases $n$43, $n$44, $n$45, and $n$46, and the non-hyperoctahedral cases are characterized by the absence of the hyperoctahedral alternative (Mang et al., 2019).

A central classification program introduces the analyzer

$n$47

where $n$48 records block sizes, $n$49 block color sums, $n$50 total color sums, $n$51 and $n$52 certain color distances between consecutive legs of a block, and $n$53 color distances between crossing blocks (Mang et al., 2020). The main theorem of the second classification paper shows that every non-hyperoctahedral category has parameter tuple $n$54 in the admissible parameter set $n$55, completing the necessity direction of the classification scheme begun in Part I (Mang et al., 2020, Mang et al., 2019).

Globally colorized categories form a more symmetric subclass. Such a category is invariant under global permutation of colors, and its classification is determined by the degree of reflection $n$56 together with an underlying non-colored category (Gromada, 2018). The associated unitary easy quantum groups are obtained from orthogonal easy quantum groups by tensor complexification: $n$57 with the case $n$58 replaced by $n$59 (Gromada, 2018). A plausible implication is that the phrase “two-color partitions” names two parallel theories: one arithmetic and $n$60-hypergeometric, the other categorical and quantum-algebraic, linked less by direct theorem transfer than by a shared combinatorics of coloring and partition structure.