Ward Numbers Overview

• Ward numbers are integer arrays that arise in combinatorics via a triangular recurrence linking second‐order Eulerian numbers, set partitions, and increasing Schröder trees, as well as in integrable systems as topological charges.
• They are defined through explicit recurrences and inverse-binomial relations, enabling higher-order generalizations and connections with Riordan arrays and partition transforms.
• In integrable systems, Ward numbers correspond to the second Chern number and a π₃-degree, thereby unifying bundle curvature, energy quantization, and homotopy invariants in Ward chiral models.

Searching arXiv for recent and foundational papers on Ward numbers in combinatorics and integrable systems. Ward numbers are integer arrays with two distinct established meanings. In enumerative combinatorics, they denote a classical triangular array $W(n,k)$ introduced by M. Ward in 1934 in connection with expansions in factorial bases, later linked to second-order Eulerian numbers, set partitions with block sizes $\ge 2$, total partition trees, and increasing Schröder trees; this usage also admits higher-order generalizations $W^{(\nu)}(n,k;s,t)$ with Riordan-inverse relations to generalized Eulerian numbers (G. et al., 2013, Wang et al., 21 Jul 2025, Tankosič, 6 Aug 2025). In integrable-systems theory, the same expression denotes an integer topological charge for solutions of the $(2+1)$-dimensional Ward chiral model, identified with a second Chern number on compactified minitwistor space and with a degree in $\pi_3(U(N))$ (Plansangkate, 2015). The two usages are unrelated in definition but parallel in that each packages a structurally natural integer invariant.

1. Classical combinatorial definitions

In current combinatorial usage, the classical Ward numbers are most commonly the array $W(n,k)$ defined by the triangular recurrence

$$W(n,k)=k\,W(n-1,k)+(n+k-1)\,W(n-1,k-1)\qquad(n\ge 1,\ k\ge 0),$$

with $W(0,k)=\delta_{0,k}$. Immediate consequences recorded in the recent Schröder-tree treatment are

$$W(n,0)=0\quad(n\ge 1),\qquad W(n,1)=1,\qquad W(n,n)=(2n-1)!!,$$

and

$W(n,2)=2^{n+1}-n-3$

(Wang et al., 21 Jul 2025).

A recent recurrence-based study distinguishes two triangular arrays called Ward numbers of the first and second kind. In that notation, the second kind is defined by

$\ge 2$0

with vanishing boundary values except at $\ge 2$1, and this is the same recurrence as the classical $\ge 2$2. The first kind is defined by

$\ge 2$3

again with the same boundary convention (Tankosič, 6 Aug 2025).

These arrays are triangular in the standard sense $\ge 2$4, and the second-kind recurrence is the one that underlies the classical combinatorial and Eulerian interpretations.

2. Relation to Eulerian numbers and higher-order generalizations

A central structural property of the classical Ward numbers is their inverse-binomial relation to the second-order Eulerian numbers $\ge 2$5. The relation is

$\ge 2$6

Thus the Ward numbers are obtained by a binomial transform of the second-order Eulerian numbers and conversely; the two arrays form a Riordan inverse pair (G. et al., 2013).

The higher-order framework replaces $\ge 2$7 by the $\ge 2$8-order generalized $\ge 2$9-Ward numbers

$W^{(\nu)}(n,k;s,t)$0

defined for $W^{(\nu)}(n,k;s,t)$1 and $W^{(\nu)}(n,k;s,t)$2 by

$W^{(\nu)}(n,k;s,t)$3

with $W^{(\nu)}(n,k;s,t)$4 for $W^{(\nu)}(n,k;s,t)$5 or $W^{(\nu)}(n,k;s,t)$6 (G. et al., 2013).

These generalized Ward numbers are paired with the $W^{(\nu)}(n,k;s,t)$7-order $W^{(\nu)}(n,k;s,t)$8-Eulerian numbers by

$W^{(\nu)}(n,k;s,t)$9

and inversely

$(2+1)$0

The classical Ward numbers are recovered as the specialization

$(2+1)$1

which the same paper identifies with the associated Stirling subset numbers

$(2+1)$2

in Fekete’s notation (G. et al., 2013).

A paper on increasing Schröder trees uses the same classical recurrence as its point of departure, while the generalized theory shows that Ward numbers are naturally embedded in a larger Eulerian–Riordan framework rather than constituting an isolated sequence family.

3. Combinatorial interpretations

The classical array $(2+1)$3 has several exact combinatorial realizations. Carlitz’s interpretation gives

$(2+1)$4

where $(2+1)$5 counts set partitions of $(2+1)$6 into $(2+1)$7 blocks, each of size at least $(2+1)$8. Equivalently, $(2+1)$9 counts partitions of $\pi_3(U(N))$0 into $\pi_3(U(N))$1 blocks of size $\pi_3(U(N))$2 (Wang et al., 21 Jul 2025).

The row sums

$\pi_3(U(N))$3

count total partitions of $\pi_3(U(N))$4. More finely, the number of total partition trees on $\pi_3(U(N))$5 with $\pi_3(U(N))$6 internal vertices equals $\pi_3(U(N))$7. In this model, a total partition is represented by a semi-labeled rooted tree whose leaves are labeled by the underlying set and whose internal vertices have degree at least $\pi_3(U(N))$8 (Wang et al., 21 Jul 2025).

A newer interpretation identifies $\pi_3(U(N))$9 with increasing Schröder trees. If $W(n,k)$0 denotes the number of increasing Schröder trees with $W(n,k)$1 vertices and $W(n,k)$2 blocks, then

$W(n,k)$3

with the same initial conditions as $W(n,k)$4, hence

$W(n,k)$5

The proof isolates the leaf carrying the maximum label $W(n,k)$6 and distinguishes whether the block containing it is a singleton or not; this yields exactly the two terms of Ward’s recurrence (Wang et al., 21 Jul 2025).

The same paper constructs a direct type-preserving bijection between total partition trees and increasing Schröder trees. “Type” records block sizes for Schröder trees and, equivalently, degree-minus-one data for total partition trees. This bijection complements earlier type-preserving bijections involving set partitions and Chen’s decomposition through meadows (Wang et al., 21 Jul 2025).

The higher-order theory supplies a separate family of interpretations. For general $W(n,k)$7, $W(n,k)$8 counts $W(n,k)$9-ary increasing $$W(n,k)=k\,W(n-1,k)+(n+k-1)\,W(n-1,k-1)\qquad(n\ge 1,\ k\ge 0),$$0-forests with at most $$W(n,k)=k\,W(n-1,k)+(n+k-1)\,W(n-1,k-1)\qquad(n\ge 1,\ k\ge 0),$$1 ascents and exactly $$W(n,k)=k\,W(n-1,k)+(n+k-1)\,W(n-1,k-1)\qquad(n\ge 1,\ k\ge 0),$$2 distinguished nodes chosen from the forest-level set of “leftmost-like” nodes. For a component tree $$W(n,k)=k\,W(n-1,k)+(n+k-1)\,W(n-1,k-1)\qquad(n\ge 1,\ k\ge 0),$$3, the distinguishable set is $$W(n,k)=k\,W(n-1,k)+(n+k-1)\,W(n-1,k-1)\qquad(n\ge 1,\ k\ge 0),$$4 when $$W(n,k)=k\,W(n-1,k)+(n+k-1)\,W(n-1,k-1)\qquad(n\ge 1,\ k\ge 0),$$5, and $$W(n,k)=k\,W(n-1,k)+(n+k-1)\,W(n-1,k-1)\qquad(n\ge 1,\ k\ge 0),$$6 when $$W(n,k)=k\,W(n-1,k)+(n+k-1)\,W(n-1,k-1)\qquad(n\ge 1,\ k\ge 0),$$7; in either case the total number of distinguishable nodes is $$W(n,k)=k\,W(n-1,k)+(n+k-1)\,W(n-1,k-1)\qquad(n\ge 1,\ k\ge 0),$$8 when the forest has $$W(n,k)=k\,W(n-1,k)+(n+k-1)\,W(n-1,k-1)\qquad(n\ge 1,\ k\ge 0),$$9 ascents (G. et al., 2013).

4. Transform-based variants and Ward-related arrays

Recent work organizes Ward numbers and their relatives through Peter Luschny’s $W(0,k)=\delta_{0,k}$0-Partition transform $W(0,k)=\delta_{0,k}$1. In that framework, the Ward numbers of the first and second kind are characterized by

$W(0,k)=\delta_{0,k}$2

with $W(0,k)=\delta_{0,k}$3 or $W(0,k)=\delta_{0,k}$4 as specified in the source (Tankosič, 6 Aug 2025).

Several derived arrays are defined by simple scalings:

Family	Defining relation	Feature
Varied Ward numbers	multiply by $W(0,k)=\delta_{0,k}$5	new triangular recurrences
Binomial Ward numbers	multiply by $W(0,k)=\delta_{0,k}$6	rational-coefficient triangular recurrences
Ward–Lah numbers	$W(0,k)=\delta_{0,k}$7	explicit formula $W(0,k)=\delta_{0,k}$8
Varied Ward–Lah numbers	$W(0,k)=\delta_{0,k}$9	simple EGF
Binomial Ward–Lah numbers	$$W(n,0)=0\quad(n\ge 1),\qquad W(n,1)=1,\qquad W(n,n)=(2n-1)!!,$$0	higher-order recurrence via Sister Celine’s algorithm

For Ward–Lah numbers, the explicit formula

$$W(n,0)=0\quad(n\ge 1),\qquad W(n,1)=1,\qquad W(n,n)=(2n-1)!!,$$1

implies special values

$$W(n,0)=0\quad(n\ge 1),\qquad W(n,1)=1,\qquad W(n,n)=(2n-1)!!,$$2

Their fixed-$$W(n,0)=0\quad(n\ge 1),\qquad W(n,1)=1,\qquad W(n,n)=(2n-1)!!,$$3 exponential generating function is

$$W(n,0)=0\quad(n\ge 1),\qquad W(n,1)=1,\qquad W(n,n)=(2n-1)!!,$$4

The varied Ward–Lah numbers satisfy

$$W(n,0)=0\quad(n\ge 1),\qquad W(n,1)=1,\qquad W(n,n)=(2n-1)!!,$$5

(Tankosič, 6 Aug 2025).

The unifying description given there is that all these sequences live in a “P-transform + simple scaling” universe. Within that universe, Ward numbers mediate between Stirling-like, Lah-like, and binomially rescaled arrays rather than appearing only as a single isolated triangle (Tankosič, 6 Aug 2025).

5. Ward numbers in the integrable chiral model

In twistor-theoretic work on the $$W(n,0)=0\quad(n\ge 1),\qquad W(n,1)=1,\qquad W(n,n)=(2n-1)!!,$$6-dimensional integrable chiral model, “Ward numbers” refers not to a combinatorial array but to an integer topological invariant. The basic field is a map

$$W(n,0)=0\quad(n\ge 1),\qquad W(n,1)=1,\qquad W(n,n)=(2n-1)!!,$$7

satisfying Ward’s chiral-model equation, and integrability is encoded by a Lax pair with spectral parameter $$W(n,0)=0\quad(n\ge 1),\qquad W(n,1)=1,\qquad W(n,n)=(2n-1)!!,$$8. From a solution $$W(n,0)=0\quad(n\ge 1),\qquad W(n,1)=1,\qquad W(n,n)=(2n-1)!!,$$9, one obtains an extended solution

$W(n,2)=2^{n+1}-n-3$0

satisfying the unitary reality condition

$W(n,2)=2^{n+1}-n-3$1

(Plansangkate, 2015).

For solutions obeying the stated finite-energy fall-off and trivial-scattering condition, the twistor correspondence associates a holomorphic rank-$W(n,2)=2^{n+1}-n-3$2 vector bundle

$W(n,2)=2^{n+1}-n-3$3

where $W(n,2)=2^{n+1}-n-3$4 is the fiberwise compactification of minitwistor space. The paper identifies this compactified minitwistor space with the blow-up of the cone

$W(n,2)=2^{n+1}-n-3$5

at its vertex (Plansangkate, 2015).

The second Chern number

$W(n,2)=2^{n+1}-n-3$6

is then compared with the restricted extended solution

$W(n,2)=2^{n+1}-n-3$7

which extends to a map $W(n,2)=2^{n+1}-n-3$8. Its homotopy class is

$W(n,2)=2^{n+1}-n-3$9

The main theorem proves

$\ge 2$00

Thus, in this context, the Ward number can be identified with either the second Chern number of the twistor bundle or the third homotopy class of the restricted extended solution. For time-dependent unitons, the same paper further deduces that the total energy is proportional to this integer, using prior results that the total energy is directly proportional to $\ge 2$01 (Plansangkate, 2015).

6. Scope, significance, and current directions

Within combinatorics, Ward numbers occupy a structurally dense position. They are simultaneously tied to second-order Eulerian numbers by Riordan inversion, to associated Stirling subset numbers by specialization, to set partitions without singletons, to total partition trees, to increasing Schröder trees, and, in generalized form, to $\ge 2$02-ary increasing forests with ascent constraints (G. et al., 2013, Wang et al., 21 Jul 2025). Recent transform-based work expands this ecosystem to Ward numbers of the first and second kind, varied Ward numbers, binomial Ward numbers, and several Lah-type analogues, together with triangular, horizontal, and higher-order recurrences (Tankosič, 6 Aug 2025).

Within integrable systems, the term serves a different role: it names an integer topological charge that unifies bundle-theoretic, homotopy-theoretic, and energetic data for Ward unitons. The equality

$\ge 2$03

shows that the same integer is simultaneously a Chern number on compactified minitwistor space and a $\ge 2$04-degree of the restricted extended solution (Plansangkate, 2015).

The main conceptual caution is therefore terminological rather than mathematical: “Ward numbers” designates a combinatorial triangle in one literature and a topological charge in another. In each domain, however, the term marks an invariant with unusually rich interconnections. In combinatorics, current work continues to enlarge the network of type-preserving bijections and weighted refinements; one explicitly stated open problem is to find a direct type-preserving bijection between enriched increasing Schröder trees and Schröder trees without relying on Chen’s decomposition algorithms (Wang et al., 21 Jul 2025).