MacMahon’s Notion of Conjugation Explained

• MacMahon’s notion of conjugation is defined in distinct contexts—compositions, partitions, and semigroups—each emphasizing a form of structural exchange.
• In composition theory, the conjugation swaps binary cut/join encodings, enabling bijections that underlie Fibonacci-type recurrences.
• In partition and semigroup settings, conjugation reinterprets multiplicity and cyclic rearrangements, transforming difference conditions into new identities.

Searching arXiv for recent and relevant papers on MacMahon’s notion of conjugation and closely related uses. MacMahon’s notion of conjugation is not a single uniform operation across all literatures bearing MacMahon’s name. In the contemporary arXiv record assembled here, it appears in at least three mathematically distinct forms: as a classical operation on ordinary compositions defined by complementing a binary cut/join encoding; as ordinary Ferrers-diagram conjugation organizing MacMahon-type partition identities; and as cyclic factor switching $a=uv,\ b=vu$ in semigroup theory, where it is formalized as $p$-conjugacy. The unifying theme is structural exchange: cuts with joins, multiplicity conditions with difference conditions, or ordered factors with their cyclic reversal (Hopkins, 1 Sep 2025, Russell, 2023, Araújo et al., 2015).

1. Terminological range and historical placement

The cited literature does not support a single canonical meaning of the phrase “MacMahon’s notion of conjugation.” Instead, it records several historically connected usages.

Context	Basic operation	Characterization in the literature
Ordinary compositions	Swap $J$ and $C$ in the cut/join sequence	Classical composition conjugation
Partitions	Transpose Ferrers/Young data	Ordinary partition conjugation used in MacMahon-type identities
Semigroups	$a=uv,\ b=vu$	$p$-conjugacy / primary conjugacy

In composition theory, the terminology is explicitly MacMahonian: “The language distinguishing compositions and partitions was set by MacMahon,” and the relevant conjugation is emphatically composition-theoretic rather than partition-theoretic (Hopkins, 1 Sep 2025). In semigroup theory, the same historical source survives only implicitly: the paper does not say “MacMahon conjugacy,” but identifies MacMahon’s notion with the relation later standardized as $p$-conjugacy, arising from cyclic rearrangement of words (Araújo et al., 2015).

A common misconception is that every occurrence of “MacMahon conjugation” refers to Ferrers-diagram transposition. The current literature shows otherwise. In the composition setting, the operation is a binary-string complement; in semigroup theory, it is a two-factor cyclic swap; and only in partition-theoretic MacMahon identities does the relevant operation coincide with ordinary conjugation of Ferrers data (Hopkins, 1 Sep 2025, Russell, 2023).

2. Classical composition conjugation

In "Classical Fibonacci compositions" (Hopkins, 1 Sep 2025), MacMahon’s conjugation is defined on ordinary compositions of $n$ through the cut/join encoding of a composition. A composition is viewed as a tiling of a $1\times n$ board by tiles whose lengths are its parts. Between adjacent unit cells one records $J$ if the cells are joined within the same part and $p$0 if there is a cut separating two parts. For $p$1, the cut/join sequence is

$p$2

The paper states the definition exactly as follows: “Given the cut/join sequence of a composition, a natural operation is to swap each binary choice. This produces what he calls the conjugate composition.” Thus the conjugate $p$3 of a composition $p$4 is obtained by replacing every $p$5 by $p$6 and every $p$7 by $p$8, then decoding the resulting cut/join sequence back into a composition of the same integer.

Two explicit examples show the mechanism: $p$9 For $J$0, the sequence $J$1 becomes $J$2, which decodes to $J$3. For $J$4, the sequence $J$5 becomes $J$6, which decodes to $J$7 (Hopkins, 1 Sep 2025).

The same source is careful to distinguish this operation from partition conjugation. It “does not develop Ferrers diagrams or lattice-path conjugation in the partition sense; instead, it stays entirely in the composition setting.” The operation is therefore diagrammatic in a tiling-and-binary-string sense, not in a Ferrers-diagram sense. The historical remark that $J$8 happens to coincide with reversal is explicitly described only as an observed example; conjugation does not in general equal reversal. MacMahon’s term “inverse conjugates” is reserved for pairs such as $J$9 and $C$0, not for the definition itself (Hopkins, 1 Sep 2025).

3. Fibonacci decompositions of composition classes

The same paper uses composition conjugation in an essential way to realize the Fibonacci recurrence inside the set of ordinary compositions (Hopkins, 1 Sep 2025). The formal theorem is

$C$1

where $C$2 denotes compositions of $C$3 with parts in $C$4, $C$5 denotes compositions of $C$6 with all parts at least $C$7, and $C$8 denotes compositions of $C$9 with all parts odd.

The numerical background is

$a=uv,\ b=vu$0

so that

$a=uv,\ b=vu$1

The theorem is not a literal partition of $a=uv,\ b=vu$2 into $a=uv,\ b=vu$3 and $a=uv,\ b=vu$4, because those two sets are “not necessarily disjoint sets of compositions.” Rather, it constructs disjoint images of these two families inside $a=uv,\ b=vu$5 (Hopkins, 1 Sep 2025).

Conjugation enters on the subset of compositions whose parts are all at least $a=uv,\ b=vu$6. In cut/join language, that hypothesis means there are no adjacent $a=uv,\ b=vu$7’s. After complementing $a=uv,\ b=vu$8, the conjugate sequence has no adjacent $a=uv,\ b=vu$9’s. Since adjacent $p$0’s correspond exactly to parts of size at least $p$1, the conjugate has no part exceeding $p$2, hence lies in $p$3. Because the original cut/join sequence begins and ends with $p$4, the conjugate sequence begins and ends with $p$5, so the conjugate composition begins and ends with part $p$6. The paper then defines the actual map by deleting those boundary $p$7’s and appending a final $p$8. The resulting $p$9-composition ends in $p$0 (Hopkins, 1 Sep 2025).

The companion map from $p$1 uses no conjugation: each odd part $p$2 is replaced by

$p$3

equivalently $p$4. This image always ends in $p$5. The two images inside $p$6 are therefore disjoint because a composition cannot end simultaneously in $p$7 and $p$8. The recurrence $p$9 becomes a structural decomposition of $n$0-compositions by final part, with composition conjugation furnishing the nontrivial identification of the “ending in $n$1” class (Hopkins, 1 Sep 2025).

For $n$2, the paper’s examples make the mechanism explicit. The set of compositions of $n$3 with all parts at least $n$4 is

$n$5

The map sends

$n$6

all ending in $n$7. The odd-part family

$n$8

maps to

$n$9

all ending in $1\times n$0. Their union is exactly $1\times n$1, realizing $1\times n$2 bijectively (Hopkins, 1 Sep 2025).

4. Partition conjugation in MacMahon-type identities

In "A refinement of and a companion to MacMahon's partition identity" (Russell, 2023), the relevant conjugation is ordinary partition conjugation. The paper recalls MacMahon’s theorem in the form

$1\times n$3

where $1\times n$4 counts partitions into parts congruent to $1\times n$5 or $1\times n$6, $1\times n$7 counts partitions in which no part occurs exactly once, and $1\times n$8 counts partitions in which no consecutive integers appear as parts and all parts are at least $1\times n$9.

The paper defines conjugation explicitly: the conjugate of a partition $J$0 is a partition $J$1 in which $J$2 equals the number of parts of $J$3 that are greater than or equal to $J$4. It then observes that the equality $J$5 is “relatively trivial, as can be seen by taking the conjugates of the partitions counted by $J$6 or $J$7.” The structural reason is that multiplicity $J$8 on one side becomes a difference $J$9 between adjacent part sizes on the other, while the exclusion of part $p$00 corresponds to the lower-bound condition “all parts are at least 2” (Russell, 2023).

The same paper inserts this conjugation step into a more refined bijective mechanism. Its main refinement states that

$p$01

where $p$02 counts partitions of $p$03 into parts congruent to $p$04 or $p$05 with exactly $p$06 parts congruent to $p$07 and exactly $p$08 parts congruent to $p$09, while $p$10 counts partitions of $p$11 with no consecutive integers and all parts at least $p$12, with exactly $p$13 parts congruent to $p$14 and exactly $p$15 parts congruent to $p$16 (Russell, 2023).

The bridge between modular conditions and conjugation is a theorem of Xiong and Keith. The cited lemma says: the conjugates of partitions with $p$17-alternating sum type $p$18 are precisely those partitions of $p$19-length type $p$20. In the $p$21 specialization used for MacMahon’s theorem, the proof proceeds by converting the product-side modular data into a $p$22-regular partition, applying Xiong–Keith to get a partition with bounded multiplicities and prescribed $p$23-alternating sum type, duplicating parts to reverse that type, reinserting multiples of $p$24, and only then taking the conjugate. Conjugation is thus the terminal operation converting the multiplicity condition “no part occurs exactly once” into the gap condition “no consecutive integers appear as parts” while simultaneously converting alternating-sum data into residue-count data (Russell, 2023).

The paper further extends the same architecture to Andrews’s generalization. There the forbidden multiplicities are

$p$25

and conjugation converts them into forbidden adjacent differences

$p$26

together with the condition that the smallest odd part is at least $p$27. This makes partition conjugation the exact mechanism by which multiplicity restrictions become gap restrictions in the broader MacMahon–Andrews family (Russell, 2023).

5. Multiplicity-transfer generalizations and MacMahon-type bijections

The literature also contains MacMahon-type bijections that are explicitly not presented as literal conjugation. "A note on Andrews-MacMahon theorem" (Nyirenda, 2022) is exemplary. It states MacMahon’s theorem as follows: the number of partitions of $p$28 in which odd multiplicities are greater than $p$29 is equal to the number of partitions of $p$30 in which odd parts are congruent to $p$31. It then situates its own contribution as a generalized bijection “in the spirit of” the Andrews–Ericksson–Petrov–Romik bijection, not as a new Ferrers-diagram conjugation.

For MacMahon’s theorem in AEPR form, a partition

$p$32

is treated multiplicity-by-multiplicity. Each multiplicity has a unique decomposition

$p$33

where

$p$34

The target multiplicities are then defined by

$p$35

$p$36

This forces parts congruent to $p$37 or $p$38 to disappear and leaves precisely the odd parts congruent to $p$39 (Nyirenda, 2022).

The generalized theorem of the same paper replaces the MacMahon case $p$40 by arbitrary $p$41 with $p$42 and $p$43. The source class consists of partitions in which multiplicities congruent to $p$44 are at least $p$45 for $p$46. The target class consists of partitions in which parts not divisible by $p$47 are congruent to

$p$48

The same decomposition pattern persists: $p$49 with

$p$50

In this family, the MacMahon case reappears exactly when $p$51 (Nyirenda, 2022).

A plausible implication is that modern work often treats MacMahon’s conjugation principle at two levels. On the one hand, Ferrers conjugation remains the clean explanation of how multiplicity conditions become difference conditions. On the other hand, explicit bijections for generalized identities may bypass literal conjugation and instead use residue-sensitive multiplicity splitting, with the MacMahon case recovered as a specialization (Nyirenda, 2022).

6. Semigroup $p$52-conjugacy as a MacMahonian abstraction

"Four Notions of Conjugacy for Abstract Semigroups" (Araújo et al., 2015) studies MacMahon’s notion in a different direction. Here the paper does not use the phrase “MacMahon conjugacy”; instead it uses the standard notation

$p$53

and calls it $p$54-conjugacy, or primary conjugacy before transitive closure. The exact definition is

$p$55

Its transitive closure is denoted

$p$56

The motivation is inverse-free reformulation. In a group, ordinary conjugacy $p$57 is equivalent to the existence of $p$58 with $p$59 and $p$60. In a general semigroup, the latter formula remains meaningful even when inverses do not. The paper therefore treats $p$61-conjugacy as a semigroup-friendly analogue of group conjugacy, historically connected to cyclic rearrangement of words in free semigroups (Araújo et al., 2015).

The basic formal properties are sharply different from the group case. In every semigroup, $p$62 is reflexive and symmetric, but it need not be transitive. In a free semigroup it is an equivalence relation. In a group it coincides with ordinary group conjugacy. The paper’s inclusion picture is

$p$63

and on epigroups

$p$64

The relation with $p$65-conjugacy is not uniform: the paper proves that all three possibilities occur in semigroups with zero—$p$66, $p$67, and incomparability (Araújo et al., 2015).

Several semigroup classes clarify when this MacMahonian relation behaves well.

Semigroup class	Behavior of $p$68
Groups	Coincides with ordinary conjugacy
Free semigroups	Equivalence relation
Completely regular semigroups	$p$69
Completely simple semigroups	$p$70

The paper also records strong structural extremes. Theorem 5.4 states

$p$71

and Theorem 5.10 shows that if $p$72 is a rectangular band, then $p$73 is universal; conversely, if $p$74 is universal and $p$75 contains an idempotent, then $p$76 must be a rectangular band. At the same time, Example 4.24 exhibits a concrete failure of transitivity: $p$77 In this abstract setting, MacMahon’s notion becomes a genuinely nontrivial conjugacy theory whose principal open problems concern transitivity and class-specific characterization (Araújo et al., 2015).

7. Adjacent frameworks, limitations, and recurring misunderstandings

Several nearby arXiv papers concern MacMahonian identities or MacMahonian structures without actually developing a notion of conjugation. Their role is clarificatory.

"A Weighted Words Study of MacMahon's and Russell's Modulo 6 Identities" (Uncu, 16 Feb 2026) is explicitly about MacMahon’s modulo $p$78 partition identities, but it does not define, use, or discuss “MacMahon’s notion of conjugation” in those terms. There is “no definition of a MacMahon conjugation map,” “no Ferrers-diagram conjugation procedure attributed to MacMahon,” and “no proof using conjugation.” Instead, the paper works through weighted words, colorings of parts, transition matrices, recurrence relations, shift equations, product generating functions, finite $p$79-series identities, and an overpartition reinterpretation. It is therefore relevant as structural background to MacMahon’s theorem, not as a direct source on conjugation (Uncu, 16 Feb 2026).

"The combinatorics of MacMahon's partial fractions" (Sills, 2018) is similarly indirect. Its core object is

$p$80

together with MacMahon’s partial-fraction decomposition indexed by partitions of $p$81. The paper does not define conjugate partitions, discuss Ferrers transposition, or state the formula

$p$82

Standard partition conjugation is present only as background to the usual interpretation of $p$83 as the generating function for partitions into at most $p$84 parts (Sills, 2018).

"MacMahon's statistics on higher-dimensional partitions" (Amanov et al., 2020) moves in yet another direction. It does not define a conjugation operation on $p$85-dimensional partitions, nor self-conjugacy, nor a transpose involution. The closest analogue is symmetry under permutation of the first $p$86 coordinates of the $p$87-dimensional diagram, and the authors explicitly note that “the definitions of $p$88 and weights $p$89 are symmetric in the first $p$90 coordinates and hence we may repeat the proof by ‘rotation’, i.e. moving any coordinate as the first one.” A plausible implication is that the higher-dimensional successor of plane-partition conjugation is not a unique involution but an $p$91-action by coordinate permutation (Amanov et al., 2020).

Taken together, these papers delimit the topic. MacMahon’s notion of conjugation is direct and explicit in classical composition theory, direct and structurally decisive in partition identities, and abstractly reformulated in semigroup theory. By contrast, many MacMahonian papers are about identities, generating functions, or higher-dimensional symmetries in which conjugation is only implicit, background, or altogether absent (Hopkins, 1 Sep 2025, Russell, 2023, Araújo et al., 2015).