Jacobi Permutations: Diverse Constructions

• Jacobi permutations are a collection of combinatorial objects defined in various contexts, including Lie theory, Viennot’s enumeration, and Jacobi-Stirling frameworks.
• They model universal Lie identities, pattern avoidance, and dihedral symmetries through methods like shuffle algebras, generating functions, and continued fractions.
• Applications span from algebraic identities and combinatorial enumeration to arithmetical matrix symmetries and elliptic function interpretations.

“Jacobi permutations” is not a single universally standardized notion. In contemporary literature, the label appears across several distinct constructions involving permutations, permutation families, or permutation actions: Jacobi subsets and Jacobi elements of the symmetric group that encode universal Lie identities, Viennot’s Jacobi permutations counted by the Euler numbers, Jacobi-Stirling multiset permutations and Jacobi-Stirling permutation pairs, and permutation models attached to Jacobi elliptic functions or Jacobi-type continued fractions. A further arithmetic use concerns permutation symmetries of sequential matrices whose Jacobi-symbol patterns are controlled by dihedral actions rather than by a permutation class in $S_n$ itself (Ivanov et al., 2017, Henke et al., 15 Sep 2025, Andrews et al., 2011, Pain, 16 Feb 2026, Ayub et al., 2018).

1. Terminological scope

The literature uses closely related expressions for several non-equivalent objects. The most important senses are summarized below.

Sense	Basic object	Defining feature
Lie-theoretic	subset $T\subseteq S_n$ or $\lambda\in \mathbb Z[S_n]$	universal vanishing of a permutation-sum of left-normed Lie brackets
Viennot-type	ordinary permutation $\pi$ of a finite set $S$	every rightward larger-than-$x$ consecutive block $\rho_\pi(x)$ has even length
Jacobi-Stirling	multiset permutation or permutation pair	interval condition between equal letters, or cycle-maxima constraints
Elliptic-function / continued-fraction	alternating or cycle-alternating permutations	generating functions governed by $\mathrm{sn},\mathrm{cn},\mathrm{dn}$, S-fractions, or J-fractions
Arithmetic-matrix	dihedral permutation of matrix positions	quarter-turn rotation induces a Jacobi-symbol symmetry law

In the Lie-theoretic usage, “Jacobi” refers to generalized Jacobi identities in free Lie rings; in Viennot’s usage it refers to a specific Euler-enumerated permutation class; in the Jacobi-Stirling setting it refers to objects attached to Jacobi-Stirling numbers and polynomials; and in the elliptic-function setting it refers to combinatorial models for Jacobi elliptic functions or Jacobi--Rogers structures (Alekseev et al., 2016, Ivanov et al., 2017, Henke et al., 15 Sep 2025, Lin et al., 2020, Deb et al., 2023).

2. Jacobi subsets and Jacobi elements in the symmetric group

A central algebraic meaning of “Jacobi permutations” is the theory of permutation families in $S_n$ whose associated left-normed commutator sum vanishes in every Lie ring. With

$$[x_1,x_2,\dots,x_n]:=[[\cdots[[x_1,x_2],x_3],\dots],x_n],$$

a subset $T\subseteq S_n$0 is called Jacobi if

$T\subseteq S_n$1

holds universally. The group-ring version replaces $T\subseteq S_n$2 by

$T\subseteq S_n$3

and calls $T\subseteq S_n$4 a Jacobi element when

$T\subseteq S_n$5

in every Lie ring (Ivanov et al., 2017).

This framework generalizes the familiar identities

$T\subseteq S_n$6

and

$T\subseteq S_n$7

which correspond respectively to $T\subseteq S_n$8 and $T\subseteq S_n$9. It also includes the degree-$\lambda\in \mathbb Z[S_n]$0 identity

$\lambda\in \mathbb Z[S_n]$1

coming from $\lambda\in \mathbb Z[S_n]$2 (Alekseev et al., 2016).

The algebraic classification is expressed in shuffle language. If $\lambda\in \mathbb Z[S_n]$3 denotes the set of $\lambda\in \mathbb Z[S_n]$4-shuffles with $\lambda\in \mathbb Z[S_n]$5, then $\lambda\in \mathbb Z[S_n]$6 is Jacobi if and only if for every $\lambda\in \mathbb Z[S_n]$7,

$\lambda\in \mathbb Z[S_n]$8

For subsets, this becomes the purely combinatorial criterion

$\lambda\in \mathbb Z[S_n]$9

where $\pi$0 are the even-$\pi$1 and odd-$\pi$2 shuffle families (Ivanov et al., 2017).

A complementary construction produces explicit higher Jacobi identities. For $\pi$3 with $\pi$4, the subsets $\pi$5 are Jacobi, and the family

$\pi$6

is a basis of $\pi$7, with

$\pi$8

This gives a systematic source of universal permutation-sum identities beyond the classical Jacobi identity (Alekseev et al., 2016).

3. Viennot’s Jacobi permutations

A different and now explicit combinatorial class is defined by a local parity condition on ordinary permutations. Given a permutation $\pi$9 of a finite set $S$0 and a letter $S$1 of $S$2, let $S$3 be the maximal consecutive subword immediately to the right of $S$4 whose letters are all larger than $S$5. Then $S$6 is Jacobi if

$S$7

An equivalent recursive definition states that the empty permutation is Jacobi, and if $S$8 and $S$9 with $x$0, then $x$1 is Jacobi if and only if $x$2 and $x$3 are Jacobi and $x$4 is even. Standardization preserves the class (Henke et al., 15 Sep 2025).

If $x$5 denotes the set of Jacobi permutations in $x$6, then

$x$7

where the Euler numbers are defined by

$x$8

This places Jacobi permutations alongside alternating permutations, André permutations, and simsun permutations as Euler-enumerated families (Henke et al., 15 Sep 2025).

The modern development of the subject is a full pattern-avoidance theory inside $x$9. For single forbidden patterns $\rho_\pi(x)$0, the enumerations are explicit. The classes $\rho_\pi(x)$1, $\rho_\pi(x)$2, and $\rho_\pi(x)$3 are equinumerous, with

$\rho_\pi(x)$4

for $\rho_\pi(x)$5. For $\rho_\pi(x)$6-avoidance,

$\rho_\pi(x)$7

for $\rho_\pi(x)$8-avoidance,

$\rho_\pi(x)$9

and for $\mathrm{sn},\mathrm{cn},\mathrm{dn}$0-avoidance,

$\mathrm{sn},\mathrm{cn},\mathrm{dn}$1

where $\mathrm{sn},\mathrm{cn},\mathrm{dn}$2 (Henke et al., 15 Sep 2025).

The class admits refined statistic theory. The ascent EGF is

$\mathrm{sn},\mathrm{cn},\mathrm{dn}$3

the left-to-right minima EGF is

$\mathrm{sn},\mathrm{cn},\mathrm{dn}$4

and the last-letter distribution satisfies

$\mathrm{sn},\mathrm{cn},\mathrm{dn}$5

the Entringer numbers (Henke et al., 15 Sep 2025).

Its internal structure is encoded by several auxiliary models. For $\mathrm{sn},\mathrm{cn},\mathrm{dn}$6- and $\mathrm{sn},\mathrm{cn},\mathrm{dn}$7-avoidance, the relevant trees are Jacobi trees, characterized as unlabeled binary trees in which every right subtree is of even size. For $\mathrm{sn},\mathrm{cn},\mathrm{dn}$8-avoidance, dual Jacobi trees appear. For $\mathrm{sn},\mathrm{cn},\mathrm{dn}$9-avoidance, a Krattenthaler-type bijection identifies Jacobi permutations with Dyck paths whose descents are all odd. The paper also proves that every permutation in

$S_n$0

is doubly Jacobi, meaning that both $S_n$1 and $S_n$2 are Jacobi (Henke et al., 15 Sep 2025).

4. Jacobi-Stirling permutations and Jacobi-Stirling permutation pairs

A third major family consists of multiset permutations and permutation pairs attached to Jacobi-Stirling theory. For Jacobi-Stirling permutations, one introduces

$S_n$3

with total order

$S_n$4

For $S_n$5, let $S_n$6, meaning that one unbarred copy of each element of $S_n$7 is removed. A permutation of $S_n$8 is a Jacobi-Stirling permutation if for each $S_n$9, all entries between the two occurrences of $$[x_1,x_2,\dots,x_n]:=[[\cdots[[x_1,x_2],x_3],\dots],x_n],$$0 are larger than $$[x_1,x_2,\dots,x_n]:=[[\cdots[[x_1,x_2],x_3],\dots],x_n],$$1. The corresponding set is $$[x_1,x_2,\dots,x_n]:=[[\cdots[[x_1,x_2],x_3],\dots],x_n],$$2, and

$$[x_1,x_2,\dots,x_n]:=[[\cdots[[x_1,x_2],x_3],\dots],x_n],$$3

In particular,

$$[x_1,x_2,\dots,x_n]:=[[\cdots[[x_1,x_2],x_3],\dots],x_n],$$4

the usual Stirling permutations of order $$[x_1,x_2,\dots,x_n]:=[[\cdots[[x_1,x_2],x_3],\dots],x_n],$$5 (Lin et al., 2020).

These classes sit inside the more general theory of generalized Stirling permutations. For each fixed $$[x_1,x_2,\dots,x_n]:=[[\cdots[[x_1,x_2],x_3],\dots],x_n],$$6, there is a multiplicity vector $$[x_1,x_2,\dots,x_n]:=[[\cdots[[x_1,x_2],x_3],\dots],x_n],$$7 with entries in $$[x_1,x_2,\dots,x_n]:=[[\cdots[[x_1,x_2],x_3],\dots],x_n],$$8 such that

$$[x_1,x_2,\dots,x_n]:=[[\cdots[[x_1,x_2],x_3],\dots],x_n],$$9

where

$T\subseteq S_n$00

The main consequence is that $T\subseteq S_n$01 is partial $T\subseteq S_n$02-positive. More precisely, each coefficient of $T\subseteq S_n$03 is homogeneous $T\subseteq S_n$04-positive in $T\subseteq S_n$05 and $T\subseteq S_n$06, confirming the conjecture of Ma, Ma, and Yeh. The $T\subseteq S_n$07-coefficients count Jacobi-Stirling permutations subject to the local restrictions

$T\subseteq S_n$08

together with prescribed values of $T\subseteq S_n$09 and $T\subseteq S_n$10 (Lin et al., 2020).

Related but distinct are the permutation-pair models for Jacobi-Stirling numbers of the first kind. A balanced Jacobi-Stirling permutation pair of length $T\subseteq S_n$11 is an ordered pair

$T\subseteq S_n$12

such that $T\subseteq S_n$13 has one more cycle than $T\subseteq S_n$14, the cycle maxima of $T\subseteq S_n$15 below $T\subseteq S_n$16 are exactly the cycle maxima of $T\subseteq S_n$17, and for each $T\subseteq S_n$18 that is not a cycle maximum, at least one of $T\subseteq S_n$19 and $T\subseteq S_n$20 is $T\subseteq S_n$21. The number of such pairs with exactly $T\subseteq S_n$22 cycles in $T\subseteq S_n$23 is

$T\subseteq S_n$24

An unbalanced Jacobi-Stirling permutation pair of length $T\subseteq S_n$25, defined when $T\subseteq S_n$26, is an ordered pair

$T\subseteq S_n$27

such that $T\subseteq S_n$28 has $T\subseteq S_n$29 more cycles than $T\subseteq S_n$30 and the cycle maxima of $T\subseteq S_n$31 below $T\subseteq S_n$32 are exactly the cycle maxima of $T\subseteq S_n$33. The number of such pairs with exactly $T\subseteq S_n$34 cycles in $T\subseteq S_n$35 is again

$T\subseteq S_n$36

Both models realize the recurrence

$T\subseteq S_n$37

by insertion of the element $T\subseteq S_n$38 into cycle notation (Andrews et al., 2011).

5. Jacobi elliptic functions, alternating permutations, and Jacobi--Rogers structures

Several papers use permutation classes to model Jacobi elliptic functions without defining a formal class called “Jacobi permutations.” One explicit construction weights alternating permutations by peak number. If

$T\subseteq S_n$39

and $T\subseteq S_n$40 denotes the number of peaks, then the elliptic weight is

$T\subseteq S_n$41

For odd alternating permutations $T\subseteq S_n$42 and even alternating classes

$T\subseteq S_n$43

the weighted EGFs are

$T\subseteq S_n$44

The maximal element of an odd alternating permutation is necessarily a peak, and removing it gives a canonical factorization into $T\subseteq S_n$45- and $T\subseteq S_n$46-components. On EGFs this becomes

$T\subseteq S_n$47

the combinatorial form of Jacobi’s elliptic identity (Pain, 16 Feb 2026).

A Catalan-level analogue appears for restricted permutations. The Catalan--Schett polynomials $T\subseteq S_n$48 satisfy

$T\subseteq S_n$49

and also

$T\subseteq S_n$50

Moreover, there is a bijection

$T\subseteq S_n$51

preserving left peaks, so

$T\subseteq S_n$52

This is presented as a Catalan-restricted counterpart of Dumont’s permutation interpretation of Schett polynomials related to Jacobi elliptic functions (Lin et al., 2024).

Another Jacobi-related development uses cycle-alternating permutations. A permutation is cycle-alternating if it has no cycle double rises, no cycle double falls, and no fixed points; equivalently, every index is either a cycle valley or a cycle peak. The set $T\subseteq S_n$53 satisfies

$T\subseteq S_n$54

Its multivariate enumerators admit Stieltjes-type continued fractions, and after contraction and generalization the associated generalized Stieltjes--Rogers and Jacobi--Rogers matrices are interpreted by alternating Laguerre digraphs (Deb et al., 2023).

At the broadest level, permutation master polynomials $T\subseteq S_n$55 and $T\subseteq S_n$56, refining record, cycle, crossing, nesting, fixed-point-level, and cycle-count statistics, admit Jacobi-type continued fractions of the form

$T\subseteq S_n$57

These J-fractions are obtained via Foata--Zeilberger-type and Biane-type bijections from permutations to labeled Motzkin paths (Sokal et al., 2020).

6. Permutation symmetries of sequential matrices and the Jacobi symbol

A different arithmetic use concerns permutation actions on matrix positions rather than permutation classes of letters. For

$T\subseteq S_n$58

the sequential matrix $T\subseteq S_n$59 is an $T\subseteq S_n$60 matrix over $T\subseteq S_n$61. The dihedral action $T\subseteq S_n$62 includes the quarter-turn rotation

$T\subseteq S_n$63

The key structural identity is

$T\subseteq S_n$64

in $T\subseteq S_n$65. Since $T\subseteq S_n$66, repeated rotation yields

$T\subseteq S_n$67

(Ayub et al., 2018).

If

$T\subseteq S_n$68

is completely multiplicative and is applied entrywise, then

$T\subseteq S_n$69

Specializing to the Jacobi symbol $T\subseteq S_n$70, with $T\subseteq S_n$71 even so that $T\subseteq S_n$72 is odd, gives

$T\subseteq S_n$73

Equivalently,

$T\subseteq S_n$74

Thus the Jacobi-symbol matrix is invariant under $T\subseteq S_n$75 rotation when $T\subseteq S_n$76 and negated by $T\subseteq S_n$77 rotation when $T\subseteq S_n$78. This quarter-turn law refines the more familiar identity $T\subseteq S_n$79 into a dihedral permutation symmetry (Ayub et al., 2018).

7. Comparative perspective

The various notions gathered under the heading “Jacobi permutations” are linked less by a common definition than by the mathematical source of the adjective “Jacobi.” In Lie theory, the source is the Jacobi identity; in Viennot’s class, it is the Euler-enumerated permutation family introduced in the context of Jacobi elliptic functions; in Jacobi-Stirling theory, it is the Jacobi differential expression and the associated Jacobi-Stirling numbers; in continued-fraction work, it is the emergence of Jacobi-type or Jacobi--Rogers structures; and in the arithmetic matrix setting, it is the Jacobi symbol (Ivanov et al., 2017, Henke et al., 15 Sep 2025, Andrews et al., 2011, Deb et al., 2023, Ayub et al., 2018).

These objects are therefore not interchangeable. Jacobi subsets are subsets of $T\subseteq S_n$80 or elements of $T\subseteq S_n$81 characterized by universal Lie identities. Viennot’s Jacobi permutations are ordinary permutations subject to a local evenness condition. Jacobi-Stirling permutations are multiset permutations with interval constraints, while Jacobi-Stirling permutation pairs are cycle-structured pairs of ordinary permutations. Alternating and cycle-alternating permutations enter the subject through elliptic-function identities and Jacobi-type continued fractions, and the sequential-matrix construction studies permutation symmetries of matrix entries rather than a permutation class. The common feature is not a single combinatorial species but a recurring role for permutation structure in problems historically attached to the name Jacobi.