--- title: MacMahon’s Notion of Conjugation Explained url: https://www.emergentmind.com/topics/macmahon-s-notion-of-conjugation type: topic --- # MacMahon’s Notion of Conjugation Explained 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 [2509.04493][2307.16753][1503.00915]. ## 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 [2509.04493]. 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 [1503.00915]. 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 [2509.04493][2307.16753]. ## 2. Classical composition conjugation In "Classical Fibonacci compositions" [2509.04493], 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 \(C\) if there is a cut separating two parts. For \((3,1,1)\in C(5)\), the cut/join sequence is \[ J\ J\ C\ C. \] 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 \(c'\) of a composition \(c\) is obtained by replacing every \(J\) by \(C\) and every \(C\) by \(J\), then decoding the resulting cut/join sequence back into a composition of the same integer. Two explicit examples show the mechanism: \[ (3,1,1)^\prime=(1,1,3),\qquad (2,3)^\prime=(1,2,1,1). \] For \((3,1,1)\), the sequence \(JJCC\) becomes \(CCJJ\), which decodes to \((1,1,3)\). For \((2,3)\), the sequence \(JCJJ\) becomes \(CJCC\), which decodes to \((1,2,1,1)\) [2509.04493]. 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 \((3,1,1)^\prime=(1,1,3)\) 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 \((3,1,1)\) and \((1,1,3)\), not for the definition itself [2509.04493]. ## 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 [2509.04493]. The formal theorem is \[ \text{For each } n \ge 2,\ \text{there is a bijection } C_{12}(n) \cong C_{\hat{1}}(n)\cup C_o(n), \] where \(C_{12}(n)\) denotes compositions of \(n\) with parts in \(\{1,2\}\), \(C_{\hat{1}}(n)\) denotes compositions of \(n\) with all parts at least \(2\), and \(C_o(n)\) denotes compositions of \(n\) with all parts odd. The numerical background is \[ c_{12}(n)=F_{n+1},\qquad c_o(n)=F_n,\qquad c_{\hat{1}}(n)=F_{n-1}, \] so that \[ F_{n+1}=F_n+F_{n-1}=c_o(n)+c_{\hat{1}}(n). \] The theorem is not a literal partition of \(C(n)\) into \(C_{\hat{1}}(n)\) and \(C_o(n)\), because those two sets are “not necessarily disjoint sets of compositions.” Rather, it constructs disjoint images of these two families inside \(C_{12}(n)\) [2509.04493]. Conjugation enters on the subset of compositions whose parts are all at least \(2\). In cut/join language, that hypothesis means there are no adjacent \(C\)’s. After complementing \(C\leftrightarrow J\), the conjugate sequence has no adjacent \(J\)’s. Since adjacent \(J\)’s correspond exactly to parts of size at least \(3\), the conjugate has no part exceeding \(2\), hence lies in \(C_{12}(n)\). Because the original cut/join sequence begins and ends with \(J\), the conjugate sequence begins and ends with \(C\), so the conjugate composition begins and ends with part \(1\). The paper then defines the actual map by deleting those boundary \(1\)’s and appending a final \(2\). The resulting \(\{1,2\}\)-composition ends in \(2\) [2509.04493]. The companion map from \(C_o(n)\) uses no conjugation: each odd part \(2k+1\) is replaced by \[ (2^k,1), \] equivalently \(2k+1\mapsto (2,2,\dots,2,1)\). This image always ends in \(1\). The two images inside \(C_{12}(n)\) are therefore disjoint because a composition cannot end simultaneously in \(1\) and \(2\). The recurrence \(F_{n+1}=F_n+F_{n-1}\) becomes a structural decomposition of \(\{1,2\}\)-compositions by final part, with composition conjugation furnishing the nontrivial identification of the “ending in \(2\)” class [2509.04493]. For \(n=5\), the paper’s examples make the mechanism explicit. The set of compositions of \(5\) with all parts at least \(2\) is \[ \{(5),(3,2),(2,3)\}. \] The map sends \[ (5)\mapsto (1,1,1,2),\qquad (3,2)\mapsto (1,2,2),\qquad (2,3)\mapsto (2,1,2), \] all ending in \(2\). The odd-part family \[ \{(5),(3,1,1),(1,3,1),(1,1,3),(1,1,1,1,1)\} \] maps to \[ (2,2,1),\ (2,1,1,1),\ (1,2,1,1),\ (1,1,2,1),\ (1,1,1,1,1), \] all ending in \(1\). Their union is exactly \(C_{12}(5)\), realizing \(F_6=F_5+F_4\) bijectively [2509.04493]. ## 4. Partition conjugation in MacMahon-type identities In "A refinement of and a companion to MacMahon's partition identity" [2307.16753], the relevant conjugation is ordinary partition conjugation. The paper recalls MacMahon’s theorem in the form \[ A_1(n)=A_2(n)=A_3(n), \] where \(A_1(n)\) counts partitions into parts congruent to \(0,2,3,\) or \(4 \pmod 6\), \(A_2(n)\) counts partitions in which no part occurs exactly once, and \(A_3(n)\) counts partitions in which no consecutive integers appear as parts and all parts are at least \(2\). The paper defines conjugation explicitly: the conjugate of a partition \((\lambda_1,\lambda_2,\dots,\lambda_k)\) is a partition \((\mu_1,\mu_2,\dots,\mu_j)\) in which \(\mu_i\) equals the number of parts of \(\lambda\) that are greater than or equal to \(i\). It then observes that the equality \(A_2(n)=A_3(n)\) is “relatively trivial, as can be seen by taking the conjugates of the partitions counted by \(A_2(n)\) or \(A_3(n)\).” The structural reason is that multiplicity \(1\) on one side becomes a difference \(1\) between adjacent part sizes on the other, while the exclusion of part \(1\) corresponds to the lower-bound condition “all parts are at least 2” [2307.16753]. The same paper inserts this conjugation step into a more refined bijective mechanism. Its main refinement states that \[ B_1(m_1,m_2,n)=B_2(m_1,m_2,n), \] where \(B_1(m_1,m_2,n)\) counts partitions of \(n\) into parts congruent to \(0,2,3,\) or \(4 \pmod 6\) with exactly \(m_1\) parts congruent to \(2 \pmod 6\) and exactly \(m_2\) parts congruent to \(4 \pmod 6\), while \(B_2(m_1,m_2,n)\) counts partitions of \(n\) with no consecutive integers and all parts at least \(2\), with exactly \(m_2\) parts congruent to \(1 \pmod 3\) and exactly \(m_1\) parts congruent to \(2 \pmod 3\) [2307.16753]. The bridge between modular conditions and conjugation is a theorem of Xiong and Keith. The cited lemma says: the conjugates of partitions with \(m\)-alternating sum type \((s_1,\dots,s_{m-1})\) are precisely those partitions of \(m\)-length type \((s_1,\dots,s_{m-1})\). In the \(m=3\) specialization used for MacMahon’s theorem, the proof proceeds by converting the product-side modular data into a \(3\)-regular partition, applying Xiong–Keith to get a partition with bounded multiplicities and prescribed \(3\)-alternating sum type, duplicating parts to reverse that type, reinserting multiples of \(3\), 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 [2307.16753]. The paper further extends the same architecture to Andrews’s generalization. There the forbidden multiplicities are \[ \{1,3,\dots,2r-1\}, \] and conjugation converts them into forbidden adjacent differences \[ \{1,3,\dots,2r-1\}, \] together with the condition that the smallest odd part is at least \(2r+1\). This makes partition conjugation the exact mechanism by which multiplicity restrictions become gap restrictions in the broader MacMahon–Andrews family [2307.16753]. ## 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" [2212.13926] is exemplary. It states MacMahon’s theorem as follows: the number of partitions of \(n\) in which odd multiplicities are greater than \(1\) is equal to the number of partitions of \(n\) in which odd parts are congruent to \(3 \pmod 6\). 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 \[ \lambda=(\ell^{h_\ell},(\ell-1)^{h_{\ell-1}},\dots,1^{h_1}) \] is treated multiplicity-by-multiplicity. Each multiplicity has a unique decomposition \[ h_i=k_i+g_i, \] where \[ k_i\in\{0,3\},\qquad g_i\in\{0,2,4,6,8,\dots\}. \] The target multiplicities are then defined by \[ d_{6t+1}=d_{6t+5}=0,\qquad d_{6t+2}=g_{3t+1},\qquad d_{6t+4}=g_{3t+2}, \] \[ d_{6t+3}=k_{2t+1}+g_{6t+3},\qquad d_{6t+6}=k_{2t+2}+g_{6t+6}. \] This forces parts congruent to \(1\) or \(5\pmod 6\) to disappear and leaves precisely the odd parts congruent to \(3\pmod 6\) [2212.13926]. The generalized theorem of the same paper replaces the MacMahon case \(p=2,\ a=1,\ r=1\) by arbitrary \(a,p\in\mathbb N\) with \(\gcd(a,p)=1\) and \(a