Varied Ward Numbers and Recurrences

• Varied Ward numbers are modified classical sequences via a factorial normalization that produces two distinct families (first and second kind) with clear combinatorial interpretations.
• They satisfy precise triangular recurrences with explicit rational coefficients, and the associated varied Ward-Lah numbers are defined by a simple explicit formula and generating function.
• The framework leverages Sister Celine's algorithm to derive higher-order recurrences, thereby connecting varied Ward, classical Ward, and Lah number families in unified combinatorial structures.

Searching arXiv for recent and foundational papers on varied Ward numbers and related Ward-number combinatorics. arXiv search query: "varied Ward numbers" Varied Ward numbers are integer sequences obtained from classical Ward numbers by multiplying by a factorial factor and dividing by a falling-factorial factor. In the paper "Recurrence Relations for Some Integer Sequences Related to Ward Numbers" (Tankosič, 6 Aug 2025), two families are introduced—varied Ward numbers of the first kind and of the second kind—together with a related family of varied Ward-Lah numbers. The paper gives triangular recurrence relations, horizontal recurrence relations, exponential generating functions, and recurrence relations of higher order obtained by using Sister Celine's general algorithm; it also notes that most of the sequences are entered in the On-Line Encyclopedia of Integer Sequences (Tankosič, 6 Aug 2025).

1. Definitions and nomenclature

For $n,k\in\mathbb{N}_0$ with $n\ge k$, the varied Ward numbers are defined by rescaling the corresponding Ward numbers by the factor $(2n)!/(n+k)^{\underline n}$, where

$(n+k)^{\underline n}=(n+k)(n+k-1)\dotsm(k+1).$

The paper distinguishes first-kind and second-kind variants, and introduces an analogous varied Ward-Lah family (Tankosič, 6 Aug 2025).

Family	Definition	Relation
Varied Ward numbers of the first kind	$$T_1^*(n,k)=\left\uparrow \begin{matrix} n \ k \end{matrix}\right\downarrow^{*}=\dfrac{(2n)!}{(n+k)^{\underline n}}\left\uparrow \begin{matrix} n \ k \end{matrix}\right\downarrow$$	Rescaling of the unsigned Ward number of the first kind
Varied Ward numbers of the second kind	$$T_2^*(n,k)=\left\updownarrow \begin{matrix} n \ k \end{matrix}\right\updownarrow^{*}=\dfrac{(2n)!}{(n+k)^{\underline n}}\left\updownarrow \begin{matrix} n \ k \end{matrix}\right\updownarrow$$	Rescaling of the Ward number of the second kind
Varied Ward-Lah numbers	$$\genfrac\lvert\rvert{0pt}{}{n}{k}^{*}=(2n)!\binom{n-1}{k-1}$$	Related analog of the Ward-Lah numbers

The nomenclature is explained directly in the paper: since $(n+k)^{\underline n}$ counts the number of $k$-element variations of $n$ objects, these sequences are referred to as "varied Ward numbers" (Tankosič, 6 Aug 2025). The construction is therefore not an unrelated deformation of Ward numbers, but a specific multiplicative normalization tied to variation-counting.

2. First-kind and second-kind triangular recurrences

The principal structural results are triangular recurrences for the two varied Ward families. For the first kind, the theorem is

$n\ge k$0

For the second kind, the corresponding theorem is

$n\ge k$1

Both recurrences are obtained by substituting the explicit rescaling definitions into the recurrences for the underlying Ward numbers and then simplifying coefficients (Tankosič, 6 Aug 2025).

For the first kind, the derivation uses the recurrence

$n\ge k$2

For the second kind, it uses

$n\ge k$3

The important point is that the varied sequences retain a triangular structure, but the rescaling converts the ordinary Ward recurrences into formulas with explicit rational prefactors involving $n\ge k$4, $n\ge k$5, and $n\ge k$6. This suggests that the variation factor interacts nontrivially with the native combinatorial triangle while still preserving one-step recursive computability.

3. Varied Ward-Lah numbers

A closely related family in the same paper is the varied Ward-Lah numbers. They are defined by

$n\ge k$7

and explicitly by

$n\ge k$8

using the identity

$n\ge k$9

The paper also recalls the corresponding Ward-Lah numbers

$(2n)!/(n+k)^{\underline n}$0

and states the scaling relation

$(2n)!/(n+k)^{\underline n}$1

Thus the varied Ward-Lah numbers are parallel to the varied Ward numbers: the same factorial/falling-factorial renormalization converts the original Ward-Lah family into a new triangle (Tankosič, 6 Aug 2025).

Their triangular recurrence is especially simple: $(2n)!/(n+k)^{\underline n}$2 The derivation is direct from the explicit formula, since

$(2n)!/(n+k)^{\underline n}$3

The paper further gives a horizontal recurrence valid for positive integers $(2n)!/(n+k)^{\underline n}$4 with $(2n)!/(n+k)^{\underline n}$5 and $(2n)!/(n+k)^{\underline n}$6: $(2n)!/(n+k)^{\underline n}$7 For $(2n)!/(n+k)^{\underline n}$8, this reduces to the same triangular recurrence, providing an internal consistency check (Tankosič, 6 Aug 2025).

4. Generating functions, identities, and algorithmic derivation

The paper supplies an exponential generating function for the varied Ward-Lah numbers: $(2n)!/(n+k)^{\underline n}$9 Its derivation starts from the explicit formula and rewrites the resulting series as

$(n+k)^{\underline n}=(n+k)(n+k-1)\dotsm(k+1).$0

which is then identified with the standard generating function for compositions after an index shift (Tankosič, 6 Aug 2025).

An additional identity links varied Ward-Lah numbers to Lah numbers. If

$(n+k)^{\underline n}=(n+k)(n+k-1)\dotsm(k+1).$1

then the paper gives

$(n+k)^{\underline n}=(n+k)(n+k-1)\dotsm(k+1).$2

where $(n+k)^{\underline n}=(n+k)(n+k-1)\dotsm(k+1).$3 is the rising factorial. This identity expresses a scaled Lah number as a sum expansion in varied Ward-Lah numbers (Tankosič, 6 Aug 2025).

The algorithmic dimension is emphasized through Sister Celine's general algorithm. The paper explicitly states that for varied Ward-Lah numbers the algorithm reproduces the two-step triangular recurrence above, and more generally that Sister Celine's algorithm is used to derive higher-order recurrences for related sequences. A plausible implication is that the varied Ward framework is particularly compatible with symbolic recurrence discovery and automated proof techniques, not only with ad hoc manipulations.

5. Placement within the Ward-number family

Varied Ward numbers sit inside a broader combinatorial literature on Ward numbers. Generalized Ward numbers were defined earlier as the unique solution of

$(n+k)^{\underline n}=(n+k)(n+k-1)\dotsm(k+1).$4

with $(n+k)^{\underline n}=(n+k)(n+k-1)\dotsm(k+1).$5 for $(n+k)^{\underline n}=(n+k)(n+k-1)\dotsm(k+1).$6 or $(n+k)^{\underline n}=(n+k)(n+k-1)\dotsm(k+1).$7. That work also states that Ward numbers originally arise as an inverse to the second-order Eulerian numbers (G. et al., 2013).

For the ordinary, non-varied case, another paper defines the Ward numbers $(n+k)^{\underline n}=(n+k)(n+k-1)\dotsm(k+1).$8 by

$(n+k)^{\underline n}=(n+k)(n+k-1)\dotsm(k+1).$9

and records several combinatorial interpretations: $$T_1^*(n,k)=\left\uparrow \begin{matrix} n \ k \end{matrix}\right\downarrow^{*}=\dfrac{(2n)!}{(n+k)^{\underline n}}\left\uparrow \begin{matrix} n \ k \end{matrix}\right\downarrow$$0 counts partitions of a set of $$T_1^*(n,k)=\left\uparrow \begin{matrix} n \ k \end{matrix}\right\downarrow^{*}=\dfrac{(2n)!}{(n+k)^{\underline n}}\left\uparrow \begin{matrix} n \ k \end{matrix}\right\downarrow$$1 elements into $$T_1^*(n,k)=\left\uparrow \begin{matrix} n \ k \end{matrix}\right\downarrow^{*}=\dfrac{(2n)!}{(n+k)^{\underline n}}\left\uparrow \begin{matrix} n \ k \end{matrix}\right\downarrow$$2 non-singleton blocks, phylogenetic trees or total partition trees, and increasing Schröder trees with $$T_1^*(n,k)=\left\uparrow \begin{matrix} n \ k \end{matrix}\right\downarrow^{*}=\dfrac{(2n)!}{(n+k)^{\underline n}}\left\uparrow \begin{matrix} n \ k \end{matrix}\right\downarrow$$3 vertices and $$T_1^*(n,k)=\left\uparrow \begin{matrix} n \ k \end{matrix}\right\downarrow^{*}=\dfrac{(2n)!}{(n+k)^{\underline n}}\left\uparrow \begin{matrix} n \ k \end{matrix}\right\downarrow$$4 blocks (Wang et al., 21 Jul 2025).

Against that background, the defining formulas of varied Ward numbers show that they are constructed from already-established Ward triangles by a universal multiplicative factor $$T_1^*(n,k)=\left\uparrow \begin{matrix} n \ k \end{matrix}\right\downarrow^{*}=\dfrac{(2n)!}{(n+k)^{\underline n}}\left\uparrow \begin{matrix} n \ k \end{matrix}\right\downarrow$$5. The paper itself states that varied Ward numbers and varied Ward-Lah numbers have analogous relations to the classical Ward numbers, and that Sister Celine's algorithm ties these numbers to classical combinatorial triangle recurrences, confirming that these numbers are natural extensions or generalizations of existing combinatorial numbers (Tankosič, 6 Aug 2025). This suggests that varied Ward numbers are best understood not as isolated sequences, but as a renormalized branch of the Ward-number/Eulerian-number ecosystem.

6. Mathematical significance and scope

Within the source paper, the mathematical content attached to varied Ward numbers is precise and finite: explicit formulas for the first-kind and second-kind families, triangular recurrences for both, a parallel theory for varied Ward-Lah numbers, a horizontal recurrence and an exponential generating function for the latter, and an identity connecting them to Lah numbers (Tankosič, 6 Aug 2025). The emphasis is not on asymptotics or a single bijective model, but on recurrence theory, generating-function algebra, and structural relations among combinatorial triangles.

The paper's summary states that varied Ward numbers arise from multiplication of classical Ward numbers of both kinds by factors involving factorials and the number of variations of $$T_1^*(n,k)=\left\uparrow \begin{matrix} n \ k \end{matrix}\right\downarrow^{*}=\dfrac{(2n)!}{(n+k)^{\underline n}}\left\uparrow \begin{matrix} n \ k \end{matrix}\right\downarrow$$6 objects taken $$T_1^*(n,k)=\left\uparrow \begin{matrix} n \ k \end{matrix}\right\downarrow^{*}=\dfrac{(2n)!}{(n+k)^{\underline n}}\left\uparrow \begin{matrix} n \ k \end{matrix}\right\downarrow$$7 at a time, leading to rich and precise triangular recursions, horizontal recursions, and exponential generating functions. It also states that their relations to Ward, Ward-Lah, and, via certain sum identities, Lah numbers are elucidated, sometimes through nontrivial sum identities, and that higher-order recurrences are justified by Sister Celine's algorithm (Tankosič, 6 Aug 2025).

A common source of ambiguity in the literature is the term "Ward number" itself, which appears in several unrelated mathematical and physical contexts. In the present combinatorial setting, varied Ward numbers refer specifically to the rescaled Ward-number triangles defined above, not to Ward identities in quantum field theory or to topological invariants attached to Ward unitons. Within combinatorics, however, the terminology is coherent: classical Ward numbers, generalized Ward numbers, Ward-Lah numbers, varied Ward numbers, and varied Ward-Lah numbers form a connected family of recursively defined arrays tied to Eulerian-type inversions, partition structures, forests, and tree-like models [(G. et al., 2013); (Wang et al., 21 Jul 2025); (Tankosič, 6 Aug 2025)].