Kreweras Complement in Noncrossing Partitions

• Kreweras complement is a fundamental involutive transformation on noncrossing partitions defined both geometrically and algebraically, central to combinatorics and Coxeter group theory.
• It employs a unique construction that fills complementary regions in a circular diagram, facilitating analyses in free probability, RNA folding models, and cyclic sieving.
• Extensions to higher-order and m-divisible partitions reveal closed-form enumerative invariants and cyclic actions, broadening its applications across combinatorial and algebraic structures.

The Kreweras complement is a fundamental involutive transformation on the lattice of noncrossing partitions, with deep connections to Coxeter group theory, Fuss–Catalan combinatorics, free probability, promotion operators, and cyclic sieving phenomena. It admits both a geometric presentation—filling complementary regions in partition diagrams—and algebraic formulations, notably as an anti-automorphism interleaving the positions of elements in multipartite structures. The concept generalizes to higher-order and Coxeter-theoretic settings, playing a central structural role in combinatorics and its applications.

1. Classical Definition and Properties

Let $[n]=\{1,2,\ldots,n\}$ and consider $\operatorname{NCP}(n)$, the poset of noncrossing partitions of $[n]$ (ordered by refinement). The Kreweras complement $K$ is the unique map $$K: \operatorname{NCP}(n) \rightarrow \operatorname{NCP}(n)$$ with the following geometric realization: place the points $1, 2, \ldots, n$ around a circle and draw the convex hulls for each block in a given partition $\pi$. Then insert primed points $1',2',\ldots, n'$ in the gaps between consecutive integers. The Kreweras complement $K(\pi)$ is the coarsest noncrossing partition on the primed points such that the union of the diagrams for $\pi$ and $\operatorname{NCP}(n)$0 yields a maximal noncrossing partition on the $\operatorname{NCP}(n)$1-gon. This construction is unique and bijective (Shigechi, 2022, Heitsch, 2023, Ebrahimi-Fard et al., 2024).

Table: Classical Kreweras Complement (Key Properties)

Property	Statement	Reference
Anti-automorphism	$\operatorname{NCP}(n)$2	(Shigechi, 2022)
Rank-reversal	$\operatorname{NCP}(n)$3	(Shigechi, 2022)
Cyclicity	$\operatorname{NCP}(n)$4 is $\operatorname{NCP}(n)$5 rotated by $\operatorname{NCP}(n)$6 (mod $\operatorname{NCP}(n)$7); $\operatorname{NCP}(n)$8 is identity up to block rotation	(Shigechi, 2022)
Self-complementarity	Fixed points are symmetric partitions, counted by $\operatorname{NCP}(n)$9	(Shigechi, 2022)
Lattice involution	$[n]$0 is a bijection, and $[n]$1 acts as a cyclic rotation of the circle	(Shigechi, 2022)

These properties make $[n]$2 a lattice anti-automorphism and induce a dihedral action on $[n]$3 by pairing with the Simion–Ullman involution (Shigechi, 2022, Heitsch, 2023).

2. Coxeter-Theoretic and Algebraic Formulation

The Kreweras complement extends to the setting of finite Coxeter groups $[n]$4, particularly via the absolute order on $[n]$5 with respect to reflections $[n]$6, and a choice of Coxeter element $[n]$7. The noncrossing partitions relative to $[n]$8 are

$[n]$9

where $K$0 denotes the absolute order. The Kreweras complement becomes

$K$1

with periodicity $K$2, and hence $K$3 where $K$4 is the Coxeter number (Armstrong et al., 2011, Dequêne et al., 2022). In type $K$5 this recovers the geometric description above, both via permutation and combinatorial polygonal models.

Algebraically, for noncrossing partitions $K$6, $K$7 can be realized as the unique solution to

$K$8

using a partial monoid structure $K$9, based on “perfect shuffle” operations and join in the lattice of noncrossing partitions (Ebrahimi-Fard et al., 2024).

3. Cyclic Sieving, Enumerative Invariants, and Orbit Structure

The Kreweras complement generates a cyclic group action on $$K: \operatorname{NCP}(n) \rightarrow \operatorname{NCP}(n)$$0, and its orbit structure is highly constrained:

• $$K: \operatorname{NCP}(n) \rightarrow \operatorname{NCP}(n)$$1-orbits have lengths that divide $$K: \operatorname{NCP}(n) \rightarrow \operatorname{NCP}(n)$$2, related to the automorphism group of the underlying partition or associated plane tree (Heitsch, 2023).
• The number of noncrossing partitions invariant under $$K: \operatorname{NCP}(n) \rightarrow \operatorname{NCP}(n)$$3 is determined by closed-form binomial or $$K: \operatorname{NCP}(n) \rightarrow \operatorname{NCP}(n)$$4-Catalan formulas, such as the (positive) Fuss–Catalan polynomials

$$K: \operatorname{NCP}(n) \rightarrow \operatorname{NCP}(n)$$5

where $$K: \operatorname{NCP}(n) \rightarrow \operatorname{NCP}(n)$$6 are the degrees of $$K: \operatorname{NCP}(n) \rightarrow \operatorname{NCP}(n)$$7 (Krattenthaler et al., 17 Jun 2025).

This underpinning gives rise to striking cyclic sieving phenomena—evaluation of the $$K: \operatorname{NCP}(n) \rightarrow \operatorname{NCP}(n)$$8-Catalan polynomial at suitable roots of unity counts partitions fixed by a given power of $$K: \operatorname{NCP}(n) \rightarrow \operatorname{NCP}(n)$$9 (Heitsch, 2023, Krattenthaler et al., 17 Jun 2025, Armstrong et al., 2011). Fixed points correspond to symmetric noncrossing partitions, with the number given by Catalan numbers for even $1, 2, \ldots, n$0 (Shigechi, 2022).

4. Generalizations: Higher-Order and Coxeter-Theoretic Complements

Extensions of the Kreweras complement arise in various ways:

• m-divisible Noncrossing Partitions: In type $1, 2, \ldots, n$1, $1, 2, \ldots, n$2-divisible noncrossing partitions are in bijection with tuples of elements in $1, 2, \ldots, n$3 whose product is a Coxeter element, with block sizes divisible by $1, 2, \ldots, n$4. Krattenthaler–Stump’s positive Kreweras maps generalize $1, 2, \ldots, n$5, realized as pseudo-rotations, and order parameters depend on $1, 2, \ldots, n$6, $1, 2, \ldots, n$7, and the Coxeter type (Krattenthaler et al., 17 Jun 2025).
• Higher-Order Complements: For each $1, 2, \ldots, n$8, one considers $1, 2, \ldots, n$9-fold operations and $\pi$0-preserving partitions. The $\pi$1-completing Kreweras complement $\pi$2 solves $\pi$3 (Ebrahimi-Fard et al., 2024).

In these settings, the cyclic order, enumeration of fixed partitions, and cyclic sieving phenomena admit explicit closed-form descriptions.

5. Connections with Promotion, Rowmotion, and Lattice Theory

Recent developments have revealed that the Kreweras complement is closely related to classical and “toggle” actions in poset combinatorics:

• Promotion Operator: In rational and classical Catalan combinatorics, the Kreweras complement acts as Schützenberger’s promotion operator on Dyck paths and $\pi$4-chains of noncrossing partitions, with $\pi$5 corresponding to rotation and higher powers to evacuation (Shigechi, 18 Mar 2026).
• Noncrossing–Nonnesting Bijections: The complement intertwines with the Panyushev map on nonnesting partitions. For type $\pi$6, the Kreweras complement on noncrossing partitions corresponds to the Kroweras complement on nonnesting partitions under a unique support-preserving, equivariant bijection distinguished by Coxeter combinatorics and the “charmed roots” statistic (Armstrong et al., 2011, Dequêne et al., 2022).
• Partial Monoid and Incidence Coalgebras: The partial associative product and incidence coalgebra structure for noncrossing partitions are preserved under Kreweras complementation, leading to deep analogies with the arithmetic of the divisibility poset and multiplicative monoids (Ebrahimi-Fard et al., 2024).

6. Representation in Other Structures and Applications

• Plane Trees and Biological Models: The Kreweras complement corresponds to a cyclic rerooting operation on plane trees, manifesting as a shift of the root vertex. These structures model certain RNA folding patterns, and $\pi$7-orbits classify fold topologies under cyclic permutation (Heitsch, 2023).
• Noncommutative Extensions: The lattice of noncommutative crossing partitions contains the Kreweras lattice as a sublattice; the Kreweras complement extends naturally as an endomorphism in this richer category, retaining anti-automorphism properties (Shigechi, 2022).
• Free Probability: Higher-order complements, partial monoid operations, and the Möbius inversion structures underlying $\pi$8 are essential in the algebraic underpinning of relations between moments and cumulants in free probability (Ebrahimi-Fard et al., 2024).

7. Explicit Computation, Algorithms, and Examples

The Kreweras complement admits efficient algorithmic realization:

• Geometrically, by tracing regions in the circle or finding maximal noncrossing matchings for the complementary set of points.
• Via the partial monoid product, by determining admissible pairs and constructing the complement as the unique partition that “fills out” to the maximal element (Ebrahimi-Fard et al., 2024).
• In the context of Dyck paths and $\pi$9-chains, by explicit implementation of promotion operators and bijection algorithms (Shigechi, 18 Mar 2026).

Small-$1',2',\ldots, n'$0 examples (e.g., for $1',2',\ldots, n'$1 or $1',2',\ldots, n'$2) demonstrate the involutive and cyclic properties, as well as the enumeration of orbits and self-complementary partitions (Shigechi, 2022, Ebrahimi-Fard et al., 2024, Heitsch, 2023).

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References

• (Krattenthaler et al., 17 Jun 2025): Positive $1',2',\ldots, n'$3-divisible non-crossing partitions and their Kreweras maps
• (Armstrong et al., 2011): A uniform bijection between nonnesting and noncrossing partitions
• (Dequêne et al., 2022): Charmed roots and the Kroweras complement
• (Shigechi, 2022): Noncommutative crossing partitions
• (Shigechi, 18 Mar 2026): Promotion and rowmotion in rational Catalan combinatorics
• (Ebrahimi-Fard et al., 2024): Noncrossing arithmetics
• (Heitsch, 2023): Counting orbits under Kreweras complementation