Raney Extensions: Catalan and Topological Ideas

• Raney extensions are two-parameter generalizations of Catalan and Fuss–Catalan numbers defined by R₍p,r₎(k), with applications in combinatorics and random matrix theory.
• They emerge through diverse models such as tree structures, coral diagrams, and ordered partitions, connecting classical recurrences to new enumerative frameworks.
• In pointfree topology, Raney extensions refine frame–space duality by linking open sets with saturated sets to precisely characterize T₀ spaces.

Raney extensions denote several related but non-identical constructions centered on the Raney numbers

$R_{p,r}(k)=\frac{r}{pk+r}\binom{pk+r}{k},$

a two-parameter generalization of Catalan and Fuss–Catalan numbers, and on a distinct pointfree-topological notion of a Raney extension of a frame. In enumerative combinatorics, the phrase usually refers to extensions of Catalan-type recurrences, tree models, path models, and constrained partition counts indexed by $(p,r)$; in probability and random matrix theory it refers to compactly supported laws whose moments are Raney numbers; and in pointfree topology it refers to pairs $(L,C)$ that refine the frame–space adjunction so as to represent exactly the $T_0$ spaces on the spatial side (Zhou, 2015, Suarez, 2024).

1. Raney numbers as the basic extension datum

The foundational object is the Raney number

$R_{p,r}(k)=\frac{r}{pk+r}\binom{pk+r}{k},$

with the combinatorial literature typically taking $p$ to be a positive integer and $r,k$ to be nonnegative integers. The specializations

$R_{p,1}(k)=C_p(k),\qquad R_{2,1}(k)=C_k$

recover the Fuss–Catalan and classical Catalan numbers, respectively. A structural decomposition due to Hilton–Pedersen expresses Raney numbers as an $r$-fold convolution of Fuss–Catalan numbers,

$$R_{p,r}(k)=\sum_{i_1+\cdots+i_r=k}C_p(i_1)\cdots C_p(i_r),$$

and thereby places them inside the standard Catalan convolutional framework (Zhou, 2015).

A second basic feature is recurrence. For fixed $(p,r)$0, the Raney numbers satisfy

$(p,r)$1

$(p,r)$2

with initial conditions

$(p,r)$3

At $(p,r)$4, the first recurrence reduces to the classical Catalan convolution

$(p,r)$5

while at $(p,r)$6 it recovers the usual $(p,r)$7-ary-tree decomposition of the Fuss–Catalan numbers (Zhou, 2015).

The same family admits an alternative planar-embedding interpretation through coral diagrams. In that model, a $(p,r)$8-coral diagram is built from an $(p,r)$9-valent base and repeated attachment of $(L,C)$0-stars, with a forbidden leftmost attachment site that rigidifies the planar isotopy class. The number of such diagrams with exactly $(L,C)$1 attached $(L,C)$2-stars is $(L,C)$3, and an ordered-partitions identity follows: $(L,C)$4 This provides a vertical, tier-by-tier refinement complementary to the Hilton–Pedersen horizontal convolution (Beagley et al., 2015).

2. Tree models, core partitions, and classical Catalan generalization

The coral-diagram formalism gives a uniform combinatorial realization of all Raney numbers as planar embeddings of trees with one distinguished base vertex of valence $(L,C)$5 and all other internal vertices of valence $(L,C)$6. In particular, $(L,C)$7 recovers the classical interpretation of the $(L,C)$8-Catalan numbers as $(L,C)$9-ary trees with $T_0$0 internal nodes, while the identity

$T_0$1

arises from a bijection that splits the leftmost base edge and re-roots the diagram (Beagley et al., 2015).

A distinct extension arises in the theory of simultaneous core partitions. For a partition $T_0$2, the $T_0$3-set $T_0$4 consists of the hook lengths in the first column, and Anderson’s poset model states that

$T_0$5

where $T_0$6 is the set of order ideals in the poset

$T_0$7

ordered by cover relations differing by $T_0$8 or $T_0$9. For the restricted class of $R_{p,r}(k)=\frac{r}{pk+r}\binom{pk+r}{k},$0-core partitions whose parts are multiples of $R_{p,r}(k)=\frac{r}{pk+r}\binom{pk+r}{k},$1, the relevant order ideals are those in $R_{p,r}(k)=\frac{r}{pk+r}\binom{pk+r}{k},$2 satisfying the property $R_{p,r}(k)=\frac{r}{pk+r}\binom{pk+r}{k},$3: each maximal consecutive run has length divisible by $R_{p,r}(k)=\frac{r}{pk+r}\binom{pk+r}{k},$4 (Zhou, 2015).

The resulting enumeration identifies a Raney number exactly. Writing $R_{p,r}(k)=\frac{r}{pk+r}\binom{pk+r}{k},$5 with $R_{p,r}(k)=\frac{r}{pk+r}\binom{pk+r}{k},$6, the number of $R_{p,r}(k)=\frac{r}{pk+r}\binom{pk+r}{k},$7-core partitions $R_{p,r}(k)=\frac{r}{pk+r}\binom{pk+r}{k},$8 with parts that are multiples of $R_{p,r}(k)=\frac{r}{pk+r}\binom{pk+r}{k},$9 is

$p$0

Equivalently, the counting function for order ideals in $p$1 having property $p$2 satisfies the same recurrences and initial conditions as $p$3, confirming Amdeberhan’s conjecture

$p$4

under the decomposition $p$5 (Zhou, 2015).

3. Path, sequence, and cover models

Several later works extend Raney enumeration far beyond trees. One example is the theory of $p$6-threshold sequences: strictly increasing sequences $p$7 with

$p$8

These sequences are in bijection with ordered $p$9-tuples of $r,k$0-ary trees, and their total number is

$r,k$1

Proper threshold sequences, defined by $r,k$2, are counted by

$r,k$3

The same paper translates proper threshold sequences into Motzkin-like paths with arbitrarily long up steps and down steps of size at most $r,k$4, showing that these path classes are likewise enumerated by Raney numbers (Rusu, 2021).

A second path model appears for Dyck paths on column-colored lattices. If columns are colored modulo $r,k$5, a Dyck path of semilength $r,k$6 is called $r,k$7-balanced when the total number of cells below the path is equal in all residue classes. Writing

$r,k$8

the number of such paths is

$r,k$9

The proof uses a block-structure lemma forcing the column-height vector to be constant on each full $R_{p,1}(k)=C_p(k),\qquad R_{2,1}(k)=C_k$0-block, a reduction to paths below the line $R_{p,1}(k)=C_p(k),\qquad R_{2,1}(k)=C_k$1, and a cycle-lemma argument. The case $R_{p,1}(k)=C_p(k),\qquad R_{2,1}(k)=C_k$2 recovers Fried’s column-alternating enumeration, while the chessboard coloring remains outside this modulo-$R_{p,1}(k)=C_p(k),\qquad R_{2,1}(k)=C_k$3 framework and instead yields Narayana numbers (Saikia, 8 Jul 2026).

A third extension is geometric rather than lattice-path-based. On infinite $R_{p,1}(k)=C_p(k),\qquad R_{2,1}(k)=C_k$4-ary trees with constant linear segments of length $R_{p,1}(k)=C_p(k),\qquad R_{2,1}(k)=C_k$5 between successive $R_{p,1}(k)=C_p(k),\qquad R_{2,1}(k)=C_k$6-stars, a sweep-cover is a family of pairwise disjoint sibling blocks satisfying an ancestor–descendant separator condition. If $R_{p,1}(k)=C_p(k),\qquad R_{2,1}(k)=C_k$7 denotes the number of sweep-covers with $R_{p,1}(k)=C_p(k),\qquad R_{2,1}(k)=C_k$8 subcovers, then $R_{p,1}(k)=C_p(k),\qquad R_{2,1}(k)=C_k$9 and $r$0 satisfies a recurrence over integer compositions involving the auxiliary quantities

$r$1

and Stirling-number counts of non-singleton block partitions. The Raney connection is a lower bound: $r$2 and in the binary case

$r$3

exactly (Wilson, 2020).

4. Raney distributions, free probability, and random matrices

The moment formula for Raney numbers extends from combinatorics to probability. For $r$4 and $r$5, the sequence

$r$6

is positive definite, hence is the moment sequence of a compactly supported probability measure $r$7 on $r$8. For rational $r$9, the measure is absolutely continuous, with density $$R_{p,r}(k)=\sum_{i_1+\cdots+i_r=k}C_p(i_1)\cdots C_p(i_r),$$0 supported on

$$R_{p,r}(k)=\sum_{i_1+\cdots+i_r=k}C_p(i_1)\cdots C_p(i_r),$$1

and expressible in terms of Meijer $$R_{p,r}(k)=\sum_{i_1+\cdots+i_r=k}C_p(i_1)\cdots C_p(i_r),$$2-functions and generalized hypergeometric functions (Mlotkowski et al., 2012).

These measures unify several standard random-matrix limit laws. The case $$R_{p,r}(k)=\sum_{i_1+\cdots+i_r=k}C_p(i_1)\cdots C_p(i_r),$$3 gives the Fuss–Catalan distributions, which describe the limiting squared singular-value density for products of independent Ginibre matrices; $$R_{p,r}(k)=\sum_{i_1+\cdots+i_r=k}C_p(i_1)\cdots C_p(i_r),$$4, $$R_{p,r}(k)=\sum_{i_1+\cdots+i_r=k}C_p(i_1)\cdots C_p(i_r),$$5 is the Marchenko–Pastur law; and the diagonal family satisfies

$$R_{p,r}(k)=\sum_{i_1+\cdots+i_r=k}C_p(i_1)\cdots C_p(i_r),$$6

so that $$R_{p,r}(k)=\sum_{i_1+\cdots+i_r=k}C_p(i_1)\cdots C_p(i_r),$$7 is Wigner’s semicircle law centered at $$R_{p,r}(k)=\sum_{i_1+\cdots+i_r=k}C_p(i_1)\cdots C_p(i_r),$$8. In free probabilistic terms,

$$R_{p,r}(k)=\sum_{i_1+\cdots+i_r=k}C_p(i_1)\cdots C_p(i_r),$$9

so the $(p,r)$00 Raney family interpolates the multiplicative free powers of Marchenko–Pastur (Penson et al., 2011, Mlotkowski et al., 2012).

The analytic core is an algebraic equation for the Stieltjes transform. If

$(p,r)$01

then

$(p,r)$02

A trigonometric parameterization of the branch cut yields an explicit density formula on $(p,r)$03, and the edge asymptotics are explicit: for $(p,r)$04,

$(p,r)$05

while at the soft edge $(p,r)$06 the density vanishes with square-root behavior (Forrester et al., 2014).

A variational characterization is also available. For families such as $(p,r)$07, $(p,r)$08, and more generally $(p,r)$09 and $(p,r)$10, the Raney densities are the unique minimizers of logarithmic energies with one-body fields tailored to the parameters. The derivation combines Wiener–Hopf analysis of singular integral equations with the algebraic Green’s-function equation, and connects the Raney densities to Muttalib–Borodin and product-matrix ensembles (Forrester et al., 2014).

5. Dynamical, arithmetic, and algorithmic extensions

In dynamical combinatorics, the classical Raney or Calkin–Wilf tree is generalized through discrete interval exchange transformations. For a composition $(p,r)$11 and the reversal permutation $(p,r)$12, the symmetric DIET $(p,r)$13 acts on the discrete interval by blockwise translations

$(p,r)$14

A recursive function $(p,r)$15 computes the number of orbits $(p,r)$16, reduces to the Euclidean gcd recursion when $(p,r)$17, and characterizes minimality by

$(p,r)$18

The associated tree of circular compositions, rooted at $(p,r)$19, extends the classical Raney tree from coprime pairs to arbitrary composition lengths (Lapointe, 2018).

A different algorithmic extension appears in Diophantine approximation. Raney transducers are finite automata acting on $(p,r)$20-encodings of continued fractions so as to synchronize the expansions of $(p,r)$21 and $(p,r)$22. For a prime $(p,r)$23, the slow and fast transducers encode the linear fractional transformation $(p,r)$24 and support a search procedure for the bottom of the $(p,r)$25-Lagrange spectrum

$(p,r)$26

For all primes $(p,r)$27, the algorithm terminates and verifies that

$(p,r)$28

for a rational number $(p,r)$29; the largest observed values occur at the Heegner primes $(p,r)$30, $(p,r)$31, and $(p,r)$32 (Dong et al., 2024).

Another arithmetic usage of the term concerns Raney sequences in the sense of binomid indices. A finite binomid index $(p,r)$33 extends to a unique lexicographically minimal infinite binomid index $(p,r)$34, and this lex-minimal extension is eventually periodic. If $(p,r)$35 are the prefix averages and

$(p,r)$36

then the minimal period of $(p,r)$37 is

$(p,r)$38

and if $(p,r)$39 is any maximizing index then

$(p,r)$40

is a preperiod. The monoid of such lex-minimal extensions is an inductive limit of finitely presented monoids (Caalim et al., 26 May 2025).

6. Pointfree-topological Raney extensions

In a separate but now substantial line of research, a Raney extension is a pair $(p,r)$41 where $(p,r)$42 is a frame, $(p,r)$43 is a coframe, $(p,r)$44 meet-generates $(p,r)$45, and the inclusion preserves frame operations and strongly exact meets. For a topological space $(p,r)$46, the basic example is

$(p,r)$47

where $(p,r)$48 is the frame of opens and $(p,r)$49 is the coframe of saturated sets, equivalently intersections of opens. This yields an idempotent adjunction

$(p,r)$50

whose fixpoints on the space side are exactly the $(p,r)$51 spaces (Suarez, 2024, Suarez, 2024).

The spectrum $(p,r)$52 consists of the completely join-prime elements of $(p,r)$53, topologized by the basis

$(p,r)$54

For a fixed frame $(p,r)$55, the smallest and largest Raney extensions have spectra

$(p,r)$56

where $(p,r)$57 is the classical spectrum and $(p,r)$58 is the $(p,r)$59 spectrum. The topological separation axioms admit internal reformulations: sobriety corresponds to CP-compactness, the $(p,r)$60 property to $(p,r)$61-canonicity or preservation of exact meets, and the $(p,r)$62 property to Booleanity of the coframe component. A sober coreflection always exists, while a $(p,r)$63 reflection exists when morphisms are restricted to exact frame maps (Suarez, 2024).

The framework has since been embedded in a broader categorical setting. Raney extensions are one of the principal examples of spatializable $(p,r)$64-concrete categories, or SFC-categories, together with strictly zero-dimensional biframes and MT-algebras. In this setting, the induced dual adjunction is again idempotent, its Top-side fixpoints are $(p,r)$65 spaces, and fiber-initial and fiber-terminal objects recover the sober and $(p,r)$66 spaces in a uniform manner (Prezado et al., 27 Jun 2026).

Two structural refinements are especially notable. First, Raney extensions are categorically equivalent to MT-algebras equipped with Raney morphisms; the generalized Funayama envelope $(p,r)$67 provides a $(p,r)$68-hull, extending the classical Funayama envelope and $(p,r)$69-hull of frames (Bezhanishvili et al., 1 Sep 2025). Second, Raney extensions can be compared directly with strictly zero-dimensional biframes: proper subcolocales of $(p,r)$70 are order-isomorphic to essential codense subcolocales of $(p,r)$71, yielding a bijection between proper Raney extensions and essential strictly zero-dimensional biframes, although the correspondence does not extend functorially in the naive way (Suarez, 25 Sep 2025).

A recurring misconception is therefore best avoided: “Raney extension” is not a single invariant notion across mathematics. In combinatorics and probability it usually names a two-parameter extension of Catalan/Fuss–Catalan structures controlled by $(p,r)$72, whereas in pointfree topology it denotes a specific frame–coframe object designed to extend the dual of $(p,r)$73. What unifies these usages is not a common formal definition, but the systematic role of Raney’s ideas in extending one-parameter structures into richer two-parameter or structured settings.