---
title: 'Raney Extensions: Catalan and Topological Ideas'
url: https://www.emergentmind.com/topics/raney-extensions
type: topic
---

# Raney Extensions: Catalan and Topological Ideas

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 [1512.08080], [2405.02990].

## 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 [1512.08080].

A second basic feature is recurrence. For fixed $p$, the Raney numbers satisfy
\[
R_{p,1}(k)=\sum_{i=0}^{k-1}R_{p,1}(i)R_{p,p-1}(k-1-i),
\]
\[
R_{p,r}(k)=\sum_{i=0}^{k}R_{p,1}(i)R_{p,r-1}(k-i),\qquad r>1,
\]
with initial conditions
\[
R_{p,r}(0)=1,\qquad R_{p,0}(k+1)=0.
\]
At $p=2$, the first recurrence reduces to the classical Catalan convolution
\[
C_k=\sum_{i=0}^{k-1}C_i\,C_{k-1-i},
\]
while at $r=1$ it recovers the usual $p$-ary-tree decomposition of the Fuss–Catalan numbers [1512.08080].

The same family admits an alternative planar-embedding interpretation through coral diagrams. In that model, a $(p,r)$-coral diagram is built from an $(r+1)$-valent base and repeated attachment of $p$-stars, with a forbidden leftmost attachment site that rigidifies the planar isotopy class. The number of such diagrams with exactly $k$ attached $p$-stars is $R_{p,r}(k)$, and an ordered-partitions identity follows:
\[
R_{p,r}(k)=\sum_{\lambda \vdash k\ \mathrm{(ordered)}} \binom{r}{\lambda_1}\binom{p\lambda_1}{\lambda_2}\cdots \binom{p\lambda_{j-1}}{\lambda_j}.
\]
This provides a vertical, tier-by-tier refinement complementary to the Hilton–Pedersen horizontal convolution [1501.07137].

## 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 $(r+1)$ and all other internal vertices of valence $(p+1)$. In particular, $R_{p,1}(k)$ recovers the classical interpretation of the $p$-Catalan numbers as $p$-ary trees with $k$ internal nodes, while the identity
\[
R_{p,p}(k)=R_{p,1}(k+1)
\]
arises from a bijection that splits the leftmost base edge and re-roots the diagram [1501.07137].

A distinct extension arises in the theory of simultaneous core partitions. For a partition $\lambda$, the $\beta$-set $\beta(\lambda)$ consists of the hook lengths in the first column, and Anderson’s poset model states that
\[
\lambda \text{ is an }(s,t)\text{-core}\quad\iff\quad \beta(\lambda)\in J(P_{(s,t)}),
\]
where $J(P_{(s,t)})$ is the set of order ideals in the poset
\[
P_{(s,t)}=\mathbb{N}^+\setminus\{k_1s+k_2t:k_1,k_2\in\mathbb{N}\},
\]
ordered by cover relations differing by $s$ or $t$. For the restricted class of $(s,s+1)$-core partitions whose parts are multiples of $p$, the relevant order ideals are those in $T_s=P_{(s,s+1)}$ satisfying the property $P_p$: each maximal consecutive run has length divisible by $p$ [1512.08080].

The resulting enumeration identifies a Raney number exactly. Writing $s=kp+r$ with $0\le r<p$, the number of $(kp+r,kp+r+1)$-core partitions $\lambda$ with parts that are multiples of $p$ is
\[
R_{p+1,r+1}(k).
\]
Equivalently, the counting function for order ideals in $T_s$ having property $P_p$ satisfies the same recurrences and initial conditions as $R_{p+1,r+1}(k)$, confirming Amdeberhan’s conjecture
\[
\frac{s+1-p\lfloor s/p\rfloor}{s+1}\binom{s+\lfloor s/p\rfloor}{s}
=R_{p+1,r+1}(k)
\]
under the decomposition $s=kp+r$ [1512.08080].

## 3. Path, sequence, and cover models

Several later works extend Raney enumeration far beyond trees. One example is the theory of $(k,l)$-threshold sequences: strictly increasing sequences $S=(s_1,\dots,s_n)$ with
\[
ki\le s_i\le kn+l,\qquad 0\le l\le k-2.
\]
These sequences are in bijection with ordered $(l+1)$-tuples of $k$-ary trees, and their total number is
\[
R_{k,l+1}(n)=\frac{l+1}{kn+l+1}\binom{kn+l+1}{n}.
\]
Proper threshold sequences, defined by $s_n=kn+l$, are counted by
\[
R_{k,k+l}(n-1).
\]
The same paper translates proper threshold sequences into Motzkin-like paths with arbitrarily long up steps and down steps of size at most $k-1$, showing that these path classes are likewise enumerated by Raney numbers [2109.05291].

A second path model appears for Dyck paths on column-colored lattices. If columns are colored modulo $c\ge2$, a Dyck path of semilength $n$ is called $c$-balanced when the total number of cells below the path is equal in all residue classes. Writing
\[
n=cm+r-1,\qquad r\in\{1,\dots,c\},
\]
the number of such paths is
\[
R_{c+1,r}(m)=\frac{r}{(c+1)m+r}\binom{(c+1)m+r}{m}.
\]
The proof uses a block-structure lemma forcing the column-height vector to be constant on each full $c$-block, a reduction to paths below the line $y=cx+s$, and a cycle-lemma argument. The case $c=2$ recovers Fried’s column-alternating enumeration, while the chessboard coloring remains outside this modulo-$c$ framework and instead yields Narayana numbers [2607.07256].

A third extension is geometric rather than lattice-path-based. On infinite $\Delta$-ary trees with constant linear segments of length $\gamma$ between successive $\Delta$-stars, a sweep-cover is a family of pairwise disjoint sibling blocks satisfying an ancestor–descendant separator condition. If $P_{\Delta,\gamma}(n)$ denotes the number of sweep-covers with $n$ subcovers, then $P_{\Delta,\gamma}(1)=\gamma+1$ and $P_{\Delta,\gamma}(n)$ satisfies a recurrence over integer compositions involving the auxiliary quantities
\[
L_\Delta(n,k)=\sum_{X\in\Gamma_{n,k}}\prod_{x_j\in X}P_{\Delta,\gamma}(x_j)
\]
and Stirling-number counts of non-singleton block partitions. The Raney connection is a lower bound:
\[
P_{\Delta,0}(n)\ge C_{\Delta,\Delta}(n),\qquad P_{\Delta,\gamma}(n)\ge \gamma\,C_{\Delta,\Delta}(n),
\]
and in the binary case
\[
P_{2,0}(n)=C_{2,1}(n+1)
\]
exactly [2009.08549].

## 4. Raney distributions, free probability, and random matrices

The moment formula for Raney numbers extends from combinatorics to probability. For $p\ge1$ and $0<r\le p$, the sequence
\[
A_m(p,r)=\binom{mp+r}{m}\frac{r}{mp+r}
\]
is positive definite, hence is the moment sequence of a compactly supported probability measure $\mu(p,r)$ on $[0,\infty)$. For rational $p>1$, the measure is absolutely continuous, with density $W_{p,r}(x)$ supported on
\[
[0,c(p)],\qquad c(p)=p^p(p-1)^{1-p},
\]
and expressible in terms of Meijer $G$-functions and generalized hypergeometric functions [1211.7259].

These measures unify several standard random-matrix limit laws. The case $r=1$ gives the Fuss–Catalan distributions, which describe the limiting squared singular-value density for products of independent Ginibre matrices; $p=2$, $r=1$ is the Marchenko–Pastur law; and the diagonal family satisfies
\[
W_{p,p}(x)=x\,W_{p,1}(x),
\]
so that $W_{2,2}$ is Wigner’s semicircle law centered at $x=2$. In free probabilistic terms,
\[
\mu(p,1)=\mu(2,1)^{\boxtimes (p-1)},
\]
so the $r=1$ Raney family interpolates the multiplicative free powers of Marchenko–Pastur [1103.3453], [1211.7259].

The analytic core is an algebraic equation for the Stieltjes transform. If
\[
G_{p,r}(z)=\int\frac{1}{z-x}\,d\mu_{p,r}(x),\qquad w(z)=zG_{p,r}(z),
\]
then
\[
w^p-z\,w+z^r=0.
\]
A trigonometric parameterization of the branch cut yields an explicit density formula on $(0,c(p))$, and the edge asymptotics are explicit: for $r<p$,
\[
W_{p,r}(x)\sim x^{-(p-r)/p}\quad\text{as }x\downarrow0,
\]
while at the soft edge $x\uparrow c(p)$ the density vanishes with square-root behavior [1404.5759].

A variational characterization is also available. For families such as $(p,r)=(\theta+1,1)$, $(\theta/2+1,1/2)$, and more generally $(\theta/q+1,1/q)$ and $(\theta/q+1,m+1/q)$, 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 [1411.4091].

## 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 $\lambda=(\lambda_1,\dots,\lambda_r)$ and the reversal permutation $\sigma(i)=r-i+1$, the symmetric DIET $T_\lambda$ acts on the discrete interval by blockwise translations
\[
T_\lambda(x)=x+s_i,\qquad s_i=\sum_{j>i}\lambda_j-\sum_{j<i}\lambda_j,\quad x\in B_i.
\]
A recursive function $f(\lambda)$ computes the number of orbits $\gamma(T_\lambda)$, reduces to the Euclidean gcd recursion when $r=2$, and characterizes minimality by
\[
T_\lambda\text{ minimal}\iff f(\lambda)=1.
\]
The associated tree of circular compositions, rooted at $(1,1)$, extends the classical Raney tree from coprime pairs to arbitrary composition lengths [1810.13000].

A different algorithmic extension appears in Diophantine approximation. Raney transducers are finite automata acting on $L/R$-encodings of continued fractions so as to synchronize the expansions of $x$ and $px$. For a prime $p$, the slow and fast transducers encode the linear fractional transformation $\gamma(x)=px$ and support a search procedure for the bottom of the $p$-Lagrange spectrum
\[
\mathcal{L}_p=\{\lambda_p(x):x\in\mathbb{R}\},\qquad \lambda_p(x)=\max\{\lambda(x),\lambda(px)\}.
\]
For all primes $p<2000$, the algorithm terminates and verifies that
\[
\min\mathcal{L}_p=\sqrt{D_p}
\]
for a rational number $D_p$; the largest observed values occur at the Heegner primes $67$, $3$, and $163$ [2409.15480].

Another arithmetic usage of the term concerns Raney sequences in the sense of binomid indices. A finite binomid index $\eta$ extends to a unique lexicographically minimal infinite binomid index $\tilde{\eta}$, and this lex-minimal extension is eventually periodic. If $A_i=S_i/i$ are the prefix averages and
\[
A_{\max}=\max_{1\le i\le m}A_i,
\]
then the minimal period of $\tilde{\eta}$ is
\[
\gcd\{\,i\le m: A_i=A_{\max}\,\},
\]
and if $k$ is any maximizing index then
\[
k\,m(m+1)/2
\]
is a preperiod. The monoid of such lex-minimal extensions is an inductive limit of finitely presented monoids [2505.20018].

## 6. Pointfree-topological Raney extensions

In a separate but now substantial line of research, a Raney extension is a pair $(L,C)$ where $L$ is a frame, $C$ is a coframe, $L\subseteq C$ meet-generates $C$, and the inclusion preserves frame operations and strongly exact meets. For a topological space $X$, the basic example is
\[
(\Omega(X),\mathcal{U}(X)),
\]
where $\Omega(X)$ is the frame of opens and $\mathcal{U}(X)$ is the coframe of saturated sets, equivalently intersections of opens. This yields an idempotent adjunction
\[
{}_R:\mathbf{Top}\rightleftarrows \mathbf{Raney}^{op}:\mathrm{pt}_R,
\]
whose fixpoints on the space side are exactly the $T_0$ spaces [2405.02990], [2405.13437].

The spectrum $\mathrm{pt}_R(L,C)$ consists of the completely join-prime elements of $C$, topologized by the basis
\[
\varphi_{(L,C)}(a)=\{\,x\in \mathrm{pt}_R(L,C): x\le a\,\},\qquad a\in L.
\]
For a fixed frame $L$, the smallest and largest Raney extensions have spectra
\[
\mathrm{pt}_D(L)\subseteq \mathrm{pt}_R(L,C)\subseteq \mathrm{pt}(L),
\]
where $\mathrm{pt}(L)$ is the classical spectrum and $\mathrm{pt}_D(L)$ is the $T_D$ spectrum. The topological separation axioms admit internal reformulations: sobriety corresponds to CP-compactness, the $T_D$ property to $E$-canonicity or preservation of exact meets, and the $T_1$ property to Booleanity of the coframe component. A sober coreflection always exists, while a $T_D$ reflection exists when morphisms are restricted to exact frame maps [2405.13437].

The framework has since been embedded in a broader categorical setting. Raney extensions are one of the principal examples of spatializable $\mathbf{Frm}$-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 $T_0$ spaces, and fiber-initial and fiber-terminal objects recover the sober and $T_D$ spaces in a uniform manner [2606.28691].

Two structural refinements are especially notable. First, Raney extensions are categorically equivalent to MT-algebras equipped with Raney morphisms; the generalized Funayama envelope $E(R)$ provides a $T_0$-hull, extending the classical Funayama envelope and $T_D$-hull of frames [2509.01233]. Second, Raney extensions can be compared directly with strictly zero-dimensional biframes: proper subcolocales of $\mathcal{S}_o(L)$ are order-isomorphic to essential codense subcolocales of $\mathcal{S}(L)$, yielding a bijection between proper Raney extensions and essential strictly zero-dimensional biframes, although the correspondence does not extend functorially in the naive way [2509.20821].

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 $R_{p,r}$, whereas in pointfree topology it denotes a specific frame–coframe object designed to extend the dual of $\mathbf{Top}_0$. 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.

Source: https://www.emergentmind.com/topics/raney-extensions