Riordan Numbers: Paths, Arrays, and Catalan Links

• Riordan numbers are a family of combinatorial sequences with definitions ranging from lattice path enumeration (refined Motzkin paths) to entries in Riordan arrays and Catalan transforms.
• They connect classical enumeration methods with representation theory, as seen in their roles in counting hook-shaped tableaux and ballot sequences with parity constraints.
• Riordan numbers also underpin analyses using generating functions, continued fractions, and Hankel transforms, linking discrete structures to orthogonal polynomials and total positivity.

Riordan numbers are not a single uniformly defined sequence across the literature. In one prominent usage, $R(n)$ denotes the classical Riordan numbers, namely the number of Riordan paths of length $n$: Motzkin paths with no flat steps on the $x$-axis. In Riordan-array theory, however, the same phrase is also used for the little Schröder numbers, for the entries $d_{n,k}$ of a Riordan array $(g,f)$, and for Catalan-derived coefficient sequences obtained by applying Riordan operators $T(g,f)$ to $C(x)$. The subject therefore sits at the intersection of lattice-path enumeration, symmetric-group representation theory, continued fractions, Hankel transforms, and the algebra of Riordan arrays (Hemmer et al., 2 Sep 2025, Chen et al., 2016, He, 2022, Barry, 2019).

1. Terminological scope and basic formalism

The terminological ambiguity is itself part of the mathematical landscape.

Usage	Object	Representative source
Classical path-sequence usage	$R(n)$, the number of Riordan paths	(Hemmer et al., 2 Sep 2025)
Riordan-array entrywise usage	entries $d_{n,k}$ of a Riordan array $(g,f)$	(He, 2022)
Catalan-derived usage	coefficients of $n$0	(Barry, 2019)
Little-Schröder usage	the $n$1-th column of the little Schröder triangle	(Chen et al., 2016)

A Riordan array is an infinite lower-triangular matrix determined by a pair of formal power series $n$2 and $n$3, with entries

$n$4

depending on notation. The associated action on a generating function $n$5 is the fundamental theorem of Riordan arrays,

$n$6

This formalism is the common mechanism behind the different meanings of “Riordan numbers”: path-counting sequences, triangle entries, Catalan transforms, and diagonals of Riordan number triangles (Barry, 2019, He, 2022).

The same framework supports several additional structures. Horizontal recurrences are encoded by $n$7- and $n$8-sequences; vertical recurrences express $n$9 as linear combinations of the coefficients of $x$0; and diagonal analysis studies

$x$1

For ordinary Riordan triangles $x$2, the logarithmic generating function of Pascal-weighted diagonal products is

$x$3

where $x$4 and $x$5 (Lang, 2017).

2. Classical Riordan numbers as a refinement of Motzkin numbers

In the path-theoretic convention, a Riordan path of length $x$6 is a Motzkin path from $x$7 to $x$8 with steps $x$9, $d_{n,k}$0, and $d_{n,k}$1, which never goes below the $d_{n,k}$2-axis and in which flat steps $d_{n,k}$3 are forbidden on the $d_{n,k}$4-axis. The classical Riordan number $d_{n,k}$5 is the number of such paths; this is OEIS A005043. With $d_{n,k}$6 and $d_{n,k}$7, the initial values are

$d_{n,k}$8

They satisfy the well-known relation with the Motzkin numbers $d_{n,k}$9,

$(g,f)$0

and consequently have generating function

$(g,f)$1

They also admit a trinomial-coefficient description: $(g,f)$2 where $(g,f)$3 is the coefficient of $(g,f)$4 in $(g,f)$5, and $(g,f)$6 is the central trinomial coefficient (Hemmer et al., 2 Sep 2025).

This realization places Riordan numbers between the Motzkin and trinomial worlds. The identity $(g,f)$7 expresses them as a refinement of Motzkin enumeration, while the formula $(g,f)$8 makes them accessible to coefficient extraction and character-sum methods. In the 2025 representation-theoretic setting, this trinomial description is used to bridge the cases $(g,f)$9 and $T(g,f)$0 in symmetric-group character identities (Hemmer et al., 2 Sep 2025).

3. Young tableaux, ballot sequences, and character degrees

Riordan numbers have classical tableau interpretations. For $T(g,f)$1 and $T(g,f)$2, the number of Riordan paths of length $T(g,f)$3 with exactly $T(g,f)$4 flat steps and $T(g,f)$5 up steps equals the degree $T(g,f)$6 of the irreducible $T(g,f)$7-character indexed by $T(g,f)$8, where

$T(g,f)$9

Summing over $C(x)$0 gives Regev’s observation

$C(x)$1

This identifies $C(x)$2 as a degree sum over hook-like three-row shapes with two equal top rows (Hemmer et al., 2 Sep 2025).

A newer interpretation is parity-theoretic. Let

$C(x)$3

and

$C(x)$4

Then

$C(x)$5

Equivalently,

$C(x)$6

The same paper states that Riordan numbers count three-candidate ballot sequences of length $C(x)$7 in which the numbers of votes for the three candidates are all of the same parity, and that these ballot sequences are in bijection with standard Young tableaux of size $C(x)$8 having at most three rows whose row-lengths have the same parity (Hemmer et al., 2 Sep 2025).

Small cases illustrate the identity concretely. For $C(x)$9, the partitions in $R(n)$0 are $R(n)$1, $R(n)$2, and $R(n)$3, with degrees $R(n)$4, $R(n)$5, and $R(n)$6, so

$R(n)$7

On the hook-like side,

$R(n)$8

and the corresponding degree sum is again $R(n)$9 (Hemmer et al., 2 Sep 2025).

4. Character-table identities and the limits of general validity

The 2025 work places Riordan numbers inside a broader program initiated by equalities proposed by Amdeberhan for signed column sums in character tables of symmetric groups. For a partition $d_{n,k}$0, one forms a multiset $d_{n,k}$1 by doubling each part either as $d_{n,k}$2 or as two copies of $d_{n,k}$3, yielding $d_{n,k}$4 partitions of $d_{n,k}$5. The proposed identities compare signed sums over $d_{n,k}$6, consisting of partitions of $d_{n,k}$7 with at most $d_{n,k}$8 parts, all even, to unsigned sums over $d_{n,k}$9, consisting of partitions whose conjugates have only even parts (Hemmer et al., 2 Sep 2025).

These equalities are not valid in general. The summed identity holds for all $(g,f)$0 but fails at $(g,f)$1 when $(g,f)$2: the left-hand side is $(g,f)$3, while the right-hand side is $(g,f)$4. The stronger per-partition identity holds for all partitions of size $(g,f)$5 and all $(g,f)$6, but fails for some $(g,f)$7 when $(g,f)$8; the example $(g,f)$9 produces incompatible extra-column contributions $n$00 and $n$01. The paper attributes these failures to the interaction between the doubling multiset $n$02 and the parity constraints in the indexing sets (Hemmer et al., 2 Sep 2025).

Special cases nevertheless survive and are precisely where Riordan numbers reappear. If $n$03, the two indexing sets stabilize and become conjugate, and the sign twist

$n$04

forces equality. More strikingly, for rectangular partitions $n$05 and $n$06,

$n$07

Thus Riordan numbers arise as the $n$08 branch of a rectangular-partition theorem whose $n$09 branch yields central trinomial coefficients. The same paper leaves several open problems, including extending the $n$10 per-partition identity to all $n$11 and finding a direct bijection between Riordan paths and standard Young tableaux with at most three rows and equal-parity row lengths (Hemmer et al., 2 Sep 2025).

5. Riordan arrays, Schröder families, and Catalan-generated sequences

In Riordan-array theory, “Riordan numbers” often means not the Motzkin-refinement sequence $n$12, but coefficient sequences generated by Riordan operators. The 2019 paper "Generalized Catalan recurrences, Riordan arrays, elliptic curves, and orthogonal polynomials" explicitly states that it does not introduce a unique sequence called “Riordan numbers.” Instead it treats the coefficients of

$n$13

where

$n$14

is the Catalan generating function solving $n$15. In this sense, “Riordan numbers” are the coefficient sequences produced by Riordan arrays acting on $n$16, its aerated versions, or related Catalan-type series (Barry, 2019).

Two central examples are the large and little Schröder numbers. The large Schröder numbers satisfy

$n$17

together with the convolution recurrence

$n$18

The little Schröder numbers satisfy

$n$19

with recurrence

$n$20

The same paper extends this to second-, third-, and fourth-order generalized Catalan–Schröder recurrences whose generating functions are always of the form $n$21, hence $n$22 (Barry, 2019).

A different 2016 paper uses “Riordan numbers” in yet another sense: it identifies the little Schröder numbers as the $n$23-th column of the little Schröder triangle, a consistent Riordan array with $n$24. From total-positivity criteria it deduces that the little Schröder triangle is totally positive, its $n$25-th column is log-convex, and each row is log-concave. Concretely,

$n$26

and the ratio $n$27 is nondecreasing (Chen et al., 2016).

6. Recursive, vertical, dual, and diagonal theories of Riordan number triangles

The modern structural theory of Riordan numbers is recursive in several distinct senses. Horizontally, a lower-triangular matrix $n$28 is a Riordan matrix if and only if there exists a unique $n$29-sequence $n$30 such that

$n$31

and a unique $n$32-sequence such that

$n$33

These are linked to $n$34 and $n$35 by

$n$36

Vertically, the entries satisfy

$n$37

so each $n$38 is a linear combination of the coefficients of $n$39 with weights determined only by $n$40 and $n$41. The 2022 paper packages this vertical recursion into matrices $n$42, whose set forms the quasi-Riordan group $n$43 under

$n$44

It also extends the formalism to $n$45-Riordan and $n$46-Riordan arrays and illustrates it with the rook and Laguerre triangles (He, 2022).

A complementary perspective comes from reversion and duality. The 2016 paper "The Three ‘R’s and Dual Riordan Arrays" treats Riordan numbers as sequences and triangular arrays generated by Riordan arrays, and shows how series reversion yields convolutional recurrence relations. For a Riordan array $n$47, the $n$48-sequence is tied to the compositional inverse $n$49 by

$n$50

The same paper embeds $n$51 in a doubly infinite recursive matrix $n$52 and defines a dual Riordan array $n$53 by antitransposing the upper-left quadrant. This is the setting in which Patalan and super Patalan numbers are analyzed (Richardson, 2016).

Diagonal sequences provide a third mode of organization. For a Riordan triangle $n$54, the 2017 paper computes the logarithmic generating function of the Pascal–Riordan diagonal-product OGFs

$n$55

as

$n$56

where $n$57 is defined by

$n$58

and $n$59. In many examples the resulting $n$60 are rational functions with numerator triangles identified with classical arrays such as the Narayana triangle (Lang, 2017).

7. Hankel transforms, Somos-4 phenomena, orthogonal polynomials, and later extensions

Hankel transforms are central in the Catalan–Riordan framework. If a sequence $n$61 has generating function represented by a Jacobi continued fraction

$n$62

then its Hankel transform

$n$63

satisfies the Heilermann product formula

$n$64

For the large and little Schröder numbers all $n$65, so both have

$n$66

More generally, shifted generalized Catalan–Schröder families have J-fractions of the form $n$67, hence Hankel transforms $n$68. Across many examples the Hankel transforms satisfy Somos-4 recurrences

$n$69

with specific parameters tied to Riordan data and, in several worked families, to elliptic curves such as $n$70 or $n$71 (Barry, 2019).

The same 2019 paper associates orthogonal polynomials to these sequences. Every Riordan array of the form

$n$72

is the coefficient array of a family of monic orthogonal polynomials with constant three-term recurrence coefficients, and these families are modified Chebyshev polynomials of the second kind. More elaborate elliptic families produce orthogonal polynomials $n$73 whose recurrence coefficients are expressed through the coordinates of multiples $n$74 on an elliptic curve, while the corresponding moment sequence is a Riordan-generated sequence extracted from the elliptic expansion (Barry, 2019).

Later work broadens the range of derived sequences. A 2024 paper studies a family

$n$75

and shows that square symmetrizations of related Riordan arrays have principal minors equal to the Robbins numbers $n$76 and the $n$77-vertex model numbers $n$78. The paper also gives a canonical Catalan factorization, isolating the Catalan core $n$79 from a parameter-dependent deformation. Although it does not single out one preferred sequence as “the” Riordan numbers, it treats these principal-minor sequences as part of the broader Riordan-group ecology in which Catalan factorizations, symmetrizations, and integrable-lattice-model enumerations interact (Barry, 2024).

Taken together, these strands show that Riordan numbers are best understood not as a single object but as a family of related objects organized by Riordan-array methods. In one direction they are the path sequence $n$80, a refinement of the Motzkin numbers with rich representation-theoretic interpretations. In another they are triangle entries, Schröder-type columns, or Catalan transforms produced by $n$81. The unifying structures are generating functions, continued fractions, total positivity, reversion, and recursive matrix formalisms.