---
title: 'Jacobi Permutations: Diverse Constructions'
url: https://www.emergentmind.com/topics/jacobi-permutations
type: topic
---

# Jacobi Permutations: Diverse Constructions

“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 [1705.03826] [2509.11494] [1112.6111] [2602.14568] [1808.06037].

## 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 [1604.05281] [1705.03826] [2509.11494] [2005.06689] [2304.06545].

## 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 \(T\subseteq S_n\) is called Jacobi if
\[
\sum_{\sigma\in T}[x_{\sigma(1)},x_{\sigma(2)},\dots,x_{\sigma(n)}]=0
\]
holds universally. The group-ring version replaces \(T\) by
\[
\lambda=\sum_{\sigma\in S_n}\lambda(\sigma)\sigma\in\mathbb Z[S_n],
\]
and calls \(\lambda\) a Jacobi element when
\[
\sum_{\sigma\in S_n}\lambda(\sigma)[x_{\sigma(1)},\dots,x_{\sigma(n)}]=0
\]
in every Lie ring [1705.03826].

This framework generalizes the familiar identities
\[
[x_1,x_2]+[x_2,x_1]=0
\]
and
\[
[x_1,x_2,x_3]+[x_2,x_3,x_1]+[x_3,x_1,x_2]=0,
\]
which correspond respectively to \(\{12,21\}\subseteq S_2\) and \(\{123,231,312\}\subseteq S_3\). It also includes the degree-\(4\) identity
\[
[x_1,x_2,x_3,x_4]+[x_2,x_1,x_4,x_3]+[x_3,x_4,x_1,x_2]+[x_4,x_3,x_2,x_1]=0,
\]
coming from \(\{1234,2143,3412,4321\}\subseteq S_4\) [1604.05281].

The algebraic classification is expressed in shuffle language. If \(\operatorname{Sh}^1(s,t)\) denotes the set of \((s,t)\)-shuffles with \(\alpha(1)=1\), then \(\lambda\in\mathbb Z[S_n]\) is Jacobi if and only if for every \(\tau\in S_n\),
\[
\sum_{i=0}^{n-1}
\sum_{(\alpha,\beta)\in \operatorname{Sh}^1(n-i,i)}
(-1)^i\,
\lambda\!\left(
\tau\cdot
\begin{pmatrix}
1 & \cdots & i & i+1 & \cdots & n\\
\beta(i) & \cdots & \beta(1) & \alpha(1) & \cdots & \alpha(n-i)
\end{pmatrix}
\right)=0.
\]
For subsets, this becomes the purely combinatorial criterion
\[
|T\cap \tau I_n^+|=|T\cap \tau I_n^-|
\qquad (\tau\in S_n),
\]
where \(I_n^\pm\) are the even-\(i\) and odd-\(i\) shuffle families [1705.03826].

A complementary construction produces explicit higher Jacobi identities. For \(k,l\ge 1\) with \(k+l\le n\), the subsets \(T_{k,l,n}\subseteq S_n\) are Jacobi, and the family
\[
\{\mathrm{Sum}(\sigma T_{k,1,n})\mid \sigma\in S_n,\ \sigma(k+1)=1,\ 1\le k\le n-1\}
\]
is a basis of \(\ker(\beta_n)\), with
\[
\operatorname{rank}\ker(\beta_n)=(n-1)!\,(n-1).
\]
This gives a systematic source of universal permutation-sum identities beyond the classical Jacobi identity [1604.05281].

## 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 \(\pi\) of a finite set \(S\) and a letter \(x\) of \(\pi\), let \(\rho_\pi(x)\) be the maximal consecutive subword immediately to the right of \(x\) whose letters are all larger than \(x\). Then \(\pi\) is Jacobi if
\[
|\rho_\pi(x)| \text{ is even for all } x\in S.
\]
An equivalent recursive definition states that the empty permutation is Jacobi, and if \(S\neq\varnothing\) and \(\pi=\alpha y\beta\) with \(y=\min S\), then \(\pi\) is Jacobi if and only if \(\alpha\) and \(\beta\) are Jacobi and \(|\beta|\) is even. Standardization preserves the class [2509.11494].

If \(\mathfrak J_n\) denotes the set of Jacobi permutations in \(\mathfrak S_n\), then
\[
|\mathfrak J_n|=E_n,
\]
where the Euler numbers are defined by
\[
\sec x+\tan x=\sum_{n=0}^{\infty}E_n\frac{x^n}{n!}.
\]
This places Jacobi permutations alongside alternating permutations, André permutations, and simsun permutations as Euler-enumerated families [2509.11494].

The modern development of the subject is a full pattern-avoidance theory inside \(\mathfrak J_n\). For single forbidden patterns \(\sigma\in\mathfrak S_3\), the enumerations are explicit. The classes \(\mathfrak J_n(213)\), \(\mathfrak J_n(312)\), and \(\mathfrak J_n(231)\) are equinumerous, with
\[
j_{2n}(\sigma)=\frac{1}{2n+1}\binom{3n}{n},\qquad
j_{2n+1}(\sigma)=\frac{1}{2n+1}\binom{3n+1}{n+1}
\]
for \(\sigma\in\{213,312,231\}\). For \(123\)-avoidance,
\[
j_n(123)=\sum_{k=0}^{\lfloor (n-1)/2\rfloor}\frac{1}{2k+1}\binom{n-k-1}{k}\binom{n}{2k},
\]
for \(132\)-avoidance,
\[
j_n(132)=1,
\]
and for \(321\)-avoidance,
\[
j_n(321)=C_{\lfloor n/2\rfloor},
\]
where \(C_n=\frac1{n+1}\binom{2n}{n}\) [2509.11494].

The class admits refined statistic theory. The ascent EGF is
\[
J^{asc}(t,x)=1+\frac{2(e^{rx}-1)}{1+r+(r-1)e^{rx}},
\qquad r=\sqrt{1-2t},
\]
the left-to-right minima EGF is
\[
J^{lrmin}(t,x)=(\sec x+\tan x)^t,
\]
and the last-letter distribution satisfies
\[
j_{n,k}^{last}=E_{n,k},
\]
the Entringer numbers [2509.11494].

Its internal structure is encoded by several auxiliary models. For \(213\)- and \(312\)-avoidance, the relevant trees are Jacobi trees, characterized as unlabeled binary trees in which every right subtree is of even size. For \(231\)-avoidance, dual Jacobi trees appear. For \(123\)-avoidance, a Krattenthaler-type bijection identifies Jacobi permutations with Dyck paths whose descents are all odd. The paper also proves that every permutation in
\[
\mathfrak J_n(213),\quad \mathfrak J_n(231),\quad \mathfrak J_n(312)
\]
is doubly Jacobi, meaning that both \(\pi\) and \(\pi^{-1}\) are Jacobi [2509.11494].

## 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
\[
M_n:=\{\bar 1,1,1,\bar 2,2,2,\dots,\bar n,n,n\}
\]
with total order
\[
\bar 1<1<\bar 2<2<\cdots<\bar n<n.
\]
For \(S\subseteq[n]\), let \(M_{n,S}=M_n\setminus S\), meaning that one unbarred copy of each element of \(S\) is removed. A permutation of \(M_{n,S}\) is a Jacobi-Stirling permutation if for each \(i\in[n]\), all entries between the two occurrences of \(i\) are larger than \(i\). The corresponding set is \(\mathrm{JSP}_{n,S}\), and
\[
\mathrm{JSP}_{n,i}:=\bigcup_{\substack{S\subseteq[n]\\ |S|=i}}\mathrm{JSP}_{n,S}.
\]
In particular,
\[
\mathrm{JSP}_{n,n}=Q_n,
\]
the usual Stirling permutations of order \(n\) [2005.06689].

These classes sit inside the more general theory of generalized Stirling permutations. For each fixed \(S\), there is a multiplicity vector \(\mathbf m(S)\) with entries in \(\{1,2\}\) such that
\[
\mathrm{JSP}_{n,S}(x,y,z)=S_{\mathbf m(S)}(x,y,z),
\]
where
\[
S_{\mathbf m}(x,y,z)=\sum_{\pi\in Q_{\mathbf m}}
x^{\operatorname{asc}(\pi)}
y^{\operatorname{des}(\pi)}
z^{\operatorname{plat}(\pi)}.
\]
The main consequence is that \(\mathrm{JSP}_{n,i}(x,y,z)\) is partial \(\gamma\)-positive. More precisely, each coefficient of \(z^k\) is homogeneous \(\gamma\)-positive in \(x\) and \(y\), confirming the conjecture of Ma, Ma, and Yeh. The \(\gamma\)-coefficients count Jacobi-Stirling permutations subject to the local restrictions
\[
\operatorname{sddes}=0,\qquad \operatorname{fdesp}=0,
\]
together with prescribed values of \(\operatorname{mdup}\) and \(\operatorname{ascpp}\) [2005.06689].

Related but distinct are the permutation-pair models for Jacobi-Stirling numbers of the first kind. A balanced Jacobi-Stirling permutation pair of length \(n\) is an ordered pair
\[
(\pi_1,\pi_2),\qquad \pi_1\in S_{n+\gamma},\ \pi_2\in S_{n+\gamma-1},
\]
such that \(\pi_1\) has one more cycle than \(\pi_2\), the cycle maxima of \(\pi_1\) below \(n+\gamma\) are exactly the cycle maxima of \(\pi_2\), and for each \(k\) that is not a cycle maximum, at least one of \(\pi_1(k)\) and \(\pi_2(k)\) is \(\le n\). The number of such pairs with exactly \(\gamma+j\) cycles in \(\pi_1\) is
\[
\left[\begin{matrix} n\\ j\end{matrix}\right]_\gamma.
\]
An unbalanced Jacobi-Stirling permutation pair of length \(n\), defined when \(2\gamma\in\mathbb N_0\), is an ordered pair
\[
(\pi_1,\pi_2),\qquad \pi_1\in S_{n+2\gamma-1},\ \pi_2\in S_n,
\]
such that \(\pi_1\) has \(2\gamma-1\) more cycles than \(\pi_2\) and the cycle maxima of \(\pi_1\) below \(n+1\) are exactly the cycle maxima of \(\pi_2\). The number of such pairs with exactly \(j\) cycles in \(\pi_2\) is again
\[
\left[\begin{matrix} n\\ j\end{matrix}\right]_\gamma.
\]
Both models realize the recurrence
\[
\left[\begin{matrix} n\\ j\end{matrix}\right]_\gamma
=
\left[\begin{matrix} n-1\\ j-1\end{matrix}\right]_\gamma
+
(n-1)(n+2\gamma-2)
\left[\begin{matrix} n-1\\ j\end{matrix}\right]_\gamma
\]
by insertion of the element \(1\) into cycle notation [1112.6111].

## 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
\[
\pi_1<\pi_2>\pi_3<\pi_4>\cdots
\]
and \(\nu(\pi)\) denotes the number of peaks, then the elliptic weight is
\[
w(\pi)=k^{\nu(\pi)}.
\]
For odd alternating permutations \(\mathcal S_{2n+1}\) and even alternating classes
\[
\mathcal C_{2n}=\{\pi\in\mathfrak S_{2n}:\pi_1<\pi_2>\cdots>\pi_{2n}\},
\qquad
\mathcal D_{2n}=\{\pi\in\mathfrak S_{2n}:\pi_1>\pi_2<\cdots<\pi_{2n}\},
\]
the weighted EGFs are
\[
\mathrm{sn}_k(u),\qquad \mathrm{cn}_k(u),\qquad \mathrm{dn}_k(u).
\]
The maximal element of an odd alternating permutation is necessarily a peak, and removing it gives a canonical factorization into \(\mathcal C\)- and \(\mathcal D\)-components. On EGFs this becomes
\[
\mathrm{sn}_k'(u)=\mathrm{cn}_k(u)\,\mathrm{dn}_k(u),
\]
the combinatorial form of Jacobi’s elliptic identity [2602.14568].

A Catalan-level analogue appears for restricted permutations. The Catalan--Schett polynomials \(C_n(x,y)\) satisfy
\[
C_n(x,y)=\sum_{\pi\in S_n(231)} x^{\operatorname{odr}(\pi)} y^{\operatorname{oar}(\pi)}
\]
and also
\[
C_n(x,y)=\sum_{\pi\in S_n(321)} x^{\,n-2mne(\pi)} y^{\,n-2mnw(\pi^{-1})}.
\]
Moreover, there is a bijection
\[
\Psi:S_n(321)\to S_n(231)
\]
preserving left peaks, so
\[
\sum_{\pi\in S_n(321)} x^{\operatorname{lpk}_e(\pi)} y^{\operatorname{lpk}_o(\pi)}
=
\sum_{\pi\in S_n(231)} x^{\operatorname{lpk}_e(\pi)} y^{\operatorname{lpk}_o(\pi)}.
\]
This is presented as a Catalan-restricted counterpart of Dumont’s permutation interpretation of Schett polynomials related to Jacobi elliptic functions [2409.01558].

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 \(\mathfrak S_{2n}^{\rm ca}\) satisfies
\[
|\mathfrak S_{2n}^{\rm ca}|=E_{2n}.
\]
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 [2304.06545].

At the broadest level, permutation master polynomials \(Q_n\) and \(\widehat Q_n\), refining record, cycle, crossing, nesting, fixed-point-level, and cycle-count statistics, admit Jacobi-type continued fractions of the form
\[
\sum_{n=0}^\infty a_n t^n
=
\cfrac{1}{1-\gamma_0 t-\cfrac{\beta_1 t^2}{1-\gamma_1 t-\cfrac{\beta_2 t^2}{1-\cdots}}.
\]
These J-fractions are obtained via Foata--Zeilberger-type and Biane-type bijections from permutations to labeled Motzkin paths [2003.08192].

## 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
\[
m=n^2+1,
\qquad
Q_n=(a_{i,j}),\qquad
a_{i,j}=j+(i-1)n,
\]
the sequential matrix \(Q_n\) is an \(n\times n\) matrix over \(\mathbb Z/m\mathbb Z\). The dihedral action \(D_4\) includes the quarter-turn rotation
\[
\rho(A)=(a_{j,n-i+1}).
\]
The key structural identity is
\[
\rho(Q_n)=nQ_n
\]
in \(M_n(\mathbb Z/m\mathbb Z)\). Since \(n^2\equiv -1\pmod{m}\), repeated rotation yields
\[
\rho(Q_n)=nQ_n,\qquad
\rho^2(Q_n)=-Q_n,\qquad
\rho^3(Q_n)=-nQ_n
\]
[1808.06037].

If
\[
\varphi:\mathbb Z/m\mathbb Z\to\{0,1,-1\}
\]
is completely multiplicative and is applied entrywise, then
\[
\rho(\varphi(Q_n))=\varphi(n)\,\varphi(Q_n).
\]
Specializing to the Jacobi symbol \(\left(\frac{\cdot}{m}\right)\), with \(n\) even so that \(m\) is odd, gives
\[
\rho\!\left(\left(\frac{Q_n}{m}\right)\right)
=
(-1)^{n^2/4}\left(\frac{Q_n}{m}\right).
\]
Equivalently,
\[
\rho\!\left(\left(\frac{Q_n}{m}\right)\right)=
\begin{cases}
\left(\frac{Q_n}{m}\right), & n\equiv 0\pmod 4,\\[1ex]
-\left(\frac{Q_n}{m}\right), & n\equiv 2\pmod 4.
\end{cases}
\]
Thus the Jacobi-symbol matrix is invariant under \(90^\circ\) rotation when \(n\equiv 0\pmod 4\) and negated by \(90^\circ\) rotation when \(n\equiv 2\pmod 4\). This quarter-turn law refines the more familiar identity \(\left(\frac{a}{m}\right)=\left(\frac{-a}{m}\right)\) into a dihedral permutation symmetry [1808.06037].

## 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 [1705.03826] [2509.11494] [1112.6111] [2304.06545] [1808.06037].

These objects are therefore not interchangeable. Jacobi subsets are subsets of \(S_n\) or elements of \(\mathbb Z[S_n]\) 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.

Source: https://www.emergentmind.com/topics/jacobi-permutations