Non-Crossing Partitioned Permutations

• Non-crossing partitioned permutations are combinatorial structures that enforce planarity by forbidding crossings, ensuring regularity and topological realizability.
• They underpin KP soliton regularity, positroid decompositions, and moduli space geometry through explicit cyclic order constraints.
• They unify various algebraic and topological frameworks by linking decorated permutations, chord diagrams, and merging-free partitions into a coherent theory.

Searching arXiv for the cited papers and closely related work on non-crossing partitioned permutations. Non-crossing partitioned permutations are combinatorial structures in which permutation data and partition data are constrained by a cyclic or surface order so that crossings are forbidden. In the disc, this is the planar regime of classical non-crossing permutations and non-crossing partitions; in higher-topological settings it is expressed by genus $0$ conditions; and in adjacent algebraic contexts the same constraint appears through chord diagrams, decorated permutations, positroid decompositions, and $\kappa$-ordered direct sums of totally nonnegative Grassmannian data. Across these settings, non-crossingness is not merely diagrammatic: it encodes regularity, connectedness, or topological realizability (Huang et al., 2024, Ardila et al., 2013, Hock, 2023, Do et al., 2019).

1. Basic combinatorial and topological formulations

A non-crossing condition is imposed relative to an ordered ground set or to marked points on the boundary of a surface. For positroids, a non-crossing partition of $[n]$ is a partition $S_1\sqcup\cdots\sqcup S_t$ such that there do not exist $a,b,c,d$ in cyclic order with $a,c\in S_i$ and $b,d\in S_j$ for $i\neq j$; equivalently, drawing the blocks as polygons on a circle produces no crossings (Ardila et al., 2013). In the KP-soliton setting, the condition is expressed directly on permutations: two cycles $\pi^{(p)}$ and $\pi^{(q)}$ are non-crossing if the corresponding chord diagrams have no crossing chords between the two permutations (Huang et al., 2024).

The topological version packages the same idea by genus. For a permutation $\kappa$0, the genus is defined by

$\kappa$1

where $\kappa$2. For a partition, one first converts each block into a cycle by listing its elements in increasing order and then applies the same formula. At genus $\kappa$3, permutations and partitions coincide, and this is precisely the non-crossing regime (Hock, 2023).

On arbitrary compact oriented surfaces with boundary, the ambient object is a polygon diagram: a disjoint union of embedded polygons whose vertices are marked boundary points. In the disc, the oriented cyclic order around each polygon defines the cycles of a non-crossing permutation; in the annulus, it produces annular non-crossing permutations. This surface model extends the notion of non-crossing permutation beyond the disc, although on general surfaces the same abstract permutation can admit several diagram realizations (Do et al., 2019).

2. Gel'fand–Dickey reductions and regular KP solitons

For the KP hierarchy, a real KP soliton is

$\kappa$4

with $\kappa$5 built from exponential functions and a matrix $\kappa$6. In standard KP soliton theory, regularity is equivalent to total nonnegativity of $\kappa$7. Under the Gel'fand–Dickey $\kappa$8-reduction, the Lax operator satisfies

$\kappa$9

and the reduction is implemented through

$[n]$0

leading to the spectral curve

$[n]$1

For regular real solitons, one chooses the $[n]$2 distinct real roots $[n]$3, and the reduced exponentials are

$[n]$4

or, after the coordinate change used in the paper,

$[n]$5

Each $[n]$6 determines a subset $[n]$7, a matrix $[n]$8, and a derangement $[n]$9. The full soliton data is assembled by the $S_1\sqcup\cdots\sqcup S_t$0-direct sum

$S_1\sqcup\cdots\sqcup S_t$1

with columns ordered according to the sorted list of all selected roots $S_1\sqcup\cdots\sqcup S_t$2 (Huang et al., 2024).

The main combinatorial theorem states that the combined soliton is regular precisely when the individual permutations are mutually non-crossing. More explicitly, if $S_1\sqcup\cdots\sqcup S_t$3 and $S_1\sqcup\cdots\sqcup S_t$4 are not non-crossing, then the $S_1\sqcup\cdots\sqcup S_t$5-direct sum $S_1\sqcup\cdots\sqcup S_t$6 is not totally nonnegative; conversely, if the matrices are mutually non-crossing, then after suitable sign choices the $S_1\sqcup\cdots\sqcup S_t$7-direct sum becomes totally nonnegative and hence defines a regular KP soliton. For pairwise non-crossing blocks,

$S_1\sqcup\cdots\sqcup S_t$8

and the resulting soliton is regular. In this setting, non-crossingness is exactly the regularity condition, not a secondary combinatorial label (Huang et al., 2024).

The same paper gives a vertex-operator construction. With

$S_1\sqcup\cdots\sqcup S_t$9

one associates to each $a,b,c,d$0 a vertex operator $a,b,c,d$1 such that

$a,b,c,d$2

For several $a,b,c,d$3’s,

$a,b,c,d$4

A key lemma shows that if the permutations are non-crossing, then the relevant vertex-operator interaction factors are positive, so the resulting $a,b,c,d$5-function defines a regular soliton (Huang et al., 2024).

The $a,b,c,d$6-reduction yields the good Boussinesq equation. There the spectral curve is

$a,b,c,d$7

and a one-soliton associated with a pair $a,b,c,d$8 has amplitude and velocity

$a,b,c,d$9

with

$a,c\in S_i$0

The classification shows that a regular solution consists of two non-crossing families of ordinary line solitons moving in opposite directions and at most one resonant $a,c\in S_i$1-cycle, that is, at most one resonant $a,c\in S_i$2-soliton (Huang et al., 2024).

3. Positroids, decorated permutations, and non-crossing decomposition

A positroid is a matroid representable by a real full-rank matrix whose maximal minors are all nonnegative; equivalently, it is a matroid coming from the totally nonnegative part of the Grassmannian. The ordered ground set matters. The central structural theorem is that if $a,c\in S_i$3 is a positroid on $a,c\in S_i$4, then its connected components form a non-crossing partition $a,c\in S_i$5 of $a,c\in S_i$6. Conversely, if $a,c\in S_i$7 is any non-crossing partition of $a,c\in S_i$8, and $a,c\in S_i$9 is a connected positroid on $b,d\in S_j$0, then

$b,d\in S_j$1

is again a positroid. Hence every positroid is built uniquely by choosing a non-crossing partition and placing a connected positroid on each block (Ardila et al., 2013).

This decomposition has a permutation description through Postnikov’s decorated permutations. The non-crossing partition $b,d\in S_j$2 is the finest non-crossing partition of $b,d\in S_j$3 such that, for every $b,d\in S_j$4, the elements $b,d\in S_j$5 and $b,d\in S_j$6 lie in the same block. Equivalently, the blocks are the connected components of the chord diagram of $b,d\in S_j$7. If

$b,d\in S_j$8

over a non-crossing partition, then the associated decorated permutation satisfies

$b,d\in S_j$9

The paper also identifies connected positroids with stabilized-interval-free permutations: for $i\neq j$0, the number of connected positroids on $i\neq j$1 equals the number of SIF permutations on $i\neq j$2 (Ardila et al., 2013).

The same non-crossing decomposition governs polyhedral and enumerative structure. If $i\neq j$3 is a rank $i\neq j$4 positroid on $i\neq j$5, the face poset of its matroid polytope $i\neq j$6 is an induced subposet of

$i\neq j$7

where $i\neq j$8 is the poset of weighted non-crossing partitions of total weight $i\neq j$9. Enumeratively, if $\pi^{(p)}$0 is the number of positroids on $\pi^{(p)}$1 and $\pi^{(p)}$2 the number of connected positroids, then

$\pi^{(p)}$3

The free-probability interpretation identifies the moments of $\pi^{(p)}$4 with positroids and the free cumulants with connected positroids, because the moment/free-cumulant relation is governed by non-crossing partitions (Ardila et al., 2013).

4. Annular, cylindrical, and surface-generalized forms

For several boundary components, the natural topological data consists of a boundary permutation $\pi^{(p)}$5 with $\pi^{(p)}$6 cycles and a second permutation $\pi^{(p)}$7. The genus of the pair is

$\pi^{(p)}$8

This is the framework in which partitioned permutations / surfaced permutations appear, and it is the natural home of annular non-crossing permutations and partitions. The cylinder case is the first nontrivial multi-boundary instance (Hock, 2023).

For non-crossing permutations on the cylinder, the known formula is

$\pi^{(p)}$9

in the notation of the cited paper. Passing from permutations to partitions is not obtained by simply forgetting cyclic order. The reason is that a cylinder cycle with $\pi^{(q)}$0 points on one boundary and $\pi^{(q)}$1 on the other corresponds to $\pi^{(q)}$2 different permutations but only one partition block. The correction rule is therefore

$\pi^{(q)}$3

which produces the explicit formula for non-crossing partitions on the cylinder. The paper characterizes this as a precise combinatorial quotient by cyclic order, rather than a trivial identification (Hock, 2023).

The surface-theoretic generalization replaces disc and cylinder diagrams by polygon diagrams on arbitrary compact oriented surfaces with boundary. A polygon diagram on $\pi^{(q)}$4 is a disjoint union of polygons whose vertices are exactly the marked boundary points $\pi^{(q)}$5, and the count

$\pi^{(q)}$6

records equivalence classes of such diagrams on a genus-$\pi^{(q)}$7 surface with $\pi^{(q)}$8 boundary components. In the disc, the representation is unique, so $\pi^{(q)}$9 is the Catalan number. In the annulus, connected diagrams still correspond uniquely to annular non-crossing partitions/permutations, but disconnected annular permutations can admit several diagram realizations, so the count is with multiplicity (Do et al., 2019).

This broader framework shows that non-crossing permutations admit a genuine surface-topological extension. The counts satisfy pruning decompositions, explicit formulas for low-complexity surfaces, and an “almost polynomial” structure in the boundary-point variables. The leading coefficients are identified with $\kappa$00-class intersection numbers on $\kappa$01, which places non-crossing permutation theory within the geometry of moduli spaces (Do et al., 2019). This suggests that non-crossing partitioned-permutation phenomena persist well beyond planar combinatorics.

5. Constrained analogues: merging-free partitions and run-sorted permutations

A nearby but more rigid class is given by merging-free partitions. A set partition $\kappa$02 is merging-free if

$\kappa$03

Flattening deletes slashes and produces a permutation $\kappa$04. A permutation is run-sorted if its runs appear in increasing order of their minima, and flattening a merging-free partition is bijective onto run-sorted permutations. The canonical forms are restricted growth functions, and the canonical forms of merging-free partitions are characterized by the condition that every left-to-right maximum letter $\kappa$05 has at least one occurrence of $\kappa$06 to its right (Beyene et al., 2021).

The non-crossing subclass is especially rigid. If

$\kappa$07

then

$\kappa$08

and hence

$\kappa$09

with $\kappa$10. The canonical-form description states that a partition is non-crossing iff its restricted growth function is $\kappa$11-avoiding, and for merging-free partitions this is equivalent to weak unimodality: $\kappa$12 The paper explicitly notes that this theory is not the full theory of partitioned permutations in the free-probability sense of Nica–Speicher, but belongs to the same combinatorial ecosystem of partitions with compatibility constraints, non-crossing conditions, canonical encodings, and recursive decompositions (Beyene et al., 2021).

The significance of this analogue is structural rather than terminological. It exhibits how an additional local constraint, here “merging-free,” drastically reduces the size of the non-crossing class: the total count becomes $\kappa$13, much smaller than the Catalan family. A plausible implication is that many non-crossing partitioned-permutation theories are best understood as planarity constraints interacting with further algebraic or order-theoretic restrictions.

6. Conceptual role and recurrent distinctions

Several recurrent distinctions organize the subject. First, non-crossingness is not merely a visual convenience. In the KP hierarchy under Gel'fand–Dickey reductions, it is exactly the regularity criterion for the $\kappa$14-direct sum of soliton blocks (Huang et al., 2024). In positroid theory, it is exactly the constraint on connected components of the ordered ground set and the basis of the unique connected-sum classification (Ardila et al., 2013). In the genus formalism, it is exactly the genus-$\kappa$15 condition (Hock, 2023).

Second, partitions and permutations coincide only in specific regimes. On the disc, genus $\kappa$16 identifies the non-crossing partition and non-crossing permutation pictures. On the cylinder, that identification fails without correction: the passage from annular permutations to partitions requires the explicit substitution

$\kappa$17

because cyclic order within a cylinder block carries multiplicity on the permutation side (Hock, 2023). On arbitrary surfaces, the same abstract permutation can be represented by several polygon diagrams, so counting diagrams is not identical to counting permutations (Do et al., 2019).

Third, the partitioned aspect can appear at different levels. In positroids it is the non-crossing partition of connected components together with decorated permutation data. In KP solitons it is the decomposition into blocks $\kappa$18 and the product of their derangements. In surface models it is the decomposition of marked boundary points into polygon blocks together with their cyclic orders. This suggests that “non-crossing partitioned permutations” is best viewed as a family of compatible non-crossing structures rather than a single isolated formalism.

Taken together, these results position non-crossing partitioned permutations as a common language for planarity constraints in total positivity, integrable systems, topological enumeration, and combinatorial probability. The unifying principle is that an ordered or surfaced combinatorial object becomes admissible precisely when its partition structure and permutation structure can be assembled without crossings.