Ward-Lah Numbers: Factorial Binomial Arrays

• Ward-Lah numbers are a triangular factorial-binomial array defined by replacing n! with (n+k)! in the classical Lah number formula.
• They satisfy distinctive recurrences and possess an exponential generating function derived from partition-transform methods.
• This framework connects with varied and binomial Ward-Lah families, linking them to broader Lah-type literature and combinatorial models.

Searching arXiv for recent and foundational papers on Ward-Lah numbers and related Ward/Lah frameworks. I’ll look up arXiv records directly relevant to Ward-Lah numbers and their classical Lah-number background. Ward-Lah numbers are a triangular array introduced as a direct Ward-type analogue of the classical Lah numbers. In the formulation developed from Peter Luschny’s partition transform $P_n^k(\cdots)$, they are defined for $n,k\in\mathbb N_0$, $n\ge k$, by

$$\genfrac\lvert\rvert{0pt}{}{n}{k}=(-1)^k (n+k)^{\underline n}\,P_n^k(1,1,\dots),$$

and for $n\ge k\ge 1$ admit the explicit form

$$\genfrac\lvert\rvert{0pt}{}{n}{k}=\frac{(n+k)!}{k!}\binom{n-1}{k-1}.$$

The array was introduced together with varied Ward numbers and binomial Ward numbers as part of a broader study of sequences related to Ward numbers, and it is recorded in the OEIS as A357367 (Tankosič, 6 Aug 2025).

1. Definition and relation to Ward and Lah triangles

The defining framework for Ward-Lah numbers starts from two Ward-number triangles. For $n\ge k\ge 1$, the Ward numbers of the first kind satisfy

$$\left\uparrow \begin{matrix} n \ k \end{matrix}\right\downarrow = (n+k-1)\left( \left\uparrow \begin{matrix} n-1 \ k \end{matrix}\right\downarrow + \left\uparrow \begin{matrix} n-1 \ k-1 \end{matrix}\right\downarrow \right),$$

while the second-kind Ward numbers satisfy

$$\left\updownarrow \begin{matrix} n \ k \end{matrix}\right\updownarrow = k\left\updownarrow \begin{matrix} n-1 \ k \end{matrix}\right\updownarrow + (n+k-1)\left\updownarrow \begin{matrix} n-1 \ k-1 \end{matrix}\right\updownarrow.$$

In the same paper, the classical Lah numbers are recalled in the form

$$\left\lfloor \begin{matrix} n \ k \end{matrix}\right\rceil = \frac{n!}{k!}\binom{n-1}{k-1}.$$

Ward-Lah numbers are then defined by combining the Ward-number partition-transform setup with the Lah-number binomial factor $n,k\in\mathbb N_0$0 (Tankosič, 6 Aug 2025).

This construction is explicitly motivated by the identity

$n,k\in\mathbb N_0$1

which turns the partition-transform definition into the closed formula

$n,k\in\mathbb N_0$2

Compared with the classical Lah numbers,

$n,k\in\mathbb N_0$3

the Ward-Lah numbers replace $n,k\in\mathbb N_0$4 by $n,k\in\mathbb N_0$5. This is the precise sense in which the name “Ward-Lah” is used in the source: the definition parallels the Ward-number construction, while the explicit formula parallels Lah numbers (Tankosič, 6 Aug 2025).

The classical Lah numbers themselves count partitions of $n,k\in\mathbb N_0$6 into $n,k\in\mathbb N_0$7 nonempty tuples, i.e. linearly ordered blocks, and satisfy

$n,k\in\mathbb N_0$8

Ward-Lah numbers inherit the same binomial skeleton $n,k\in\mathbb N_0$9, but the available source develops them as an algebraic array rather than through a separate combinatorial counting model (Martinjak et al., 2017).

2. Explicit formulas and the Lah transform relation

The primary closed formula for Ward-Lah numbers is

$n\ge k$0

This places the array among lower-triangular factorial-binomial triangles. It also makes clear that the growth in the first index is substantially larger than for ordinary Lah numbers because of the replacement $n\ge k$1 (Tankosič, 6 Aug 2025).

A second formula expresses Ward-Lah numbers as an alternating binomial transform of Lah numbers: $n\ge k$2 Equivalently,

$n\ge k$3

This identity makes the dependence on the classical Lah triangle completely explicit: Ward-Lah numbers are not introduced as an unrelated sequence, but as a transformed Lah-type array (Tankosič, 6 Aug 2025).

This relation is especially relevant in view of the classical role of Lah numbers as connection coefficients between rising and falling factorials. For ordinary Lah numbers one has

$n\ge k$4

and several generalization programs replace the classical factorial bases by Whitney, $n\ge k$5-Whitney, multiple, or probabilistic $n\ge k$6-type systems. The Ward-Lah array belongs to that wider basis-conversion landscape, but its defining formulas are given directly through the partition transform and the alternating Lah expansion rather than through a new factorial-connection identity (Martinjak et al., 2017).

3. Recurrences and generating functions

The Ward-Lah triangle satisfies several recurrences. A triangular recurrence of Ward type is

$n\ge k$7

valid for $n\ge k$8 and $n\ge k$9. The paper also gives a second triangular recurrence with integer coefficients: $$\genfrac\lvert\rvert{0pt}{}{n}{k}=(-1)^k (n+k)^{\underline n}\,P_n^k(1,1,\dots),$$0 For $$\genfrac\lvert\rvert{0pt}{}{n}{k}=(-1)^k (n+k)^{\underline n}\,P_n^k(1,1,\dots),$$1, the horizontal recurrence specializes to

$$\genfrac\lvert\rvert{0pt}{}{n}{k}=(-1)^k (n+k)^{\underline n}\,P_n^k(1,1,\dots),$$2

These formulas are derived directly from the explicit expression and are presented as the basic recurrence structure of the array (Tankosič, 6 Aug 2025).

The general horizontal recurrence is

$$\genfrac\lvert\rvert{0pt}{}{n}{k}=(-1)^k (n+k)^{\underline n}\,P_n^k(1,1,\dots),$$3

for positive integers $$\genfrac\lvert\rvert{0pt}{}{n}{k}=(-1)^k (n+k)^{\underline n}\,P_n^k(1,1,\dots),$$4 with $$\genfrac\lvert\rvert{0pt}{}{n}{k}=(-1)^k (n+k)^{\underline n}\,P_n^k(1,1,\dots),$$5, $$\genfrac\lvert\rvert{0pt}{}{n}{k}=(-1)^k (n+k)^{\underline n}\,P_n^k(1,1,\dots),$$6, and $$\genfrac\lvert\rvert{0pt}{}{n}{k}=(-1)^k (n+k)^{\underline n}\,P_n^k(1,1,\dots),$$7. Its derivation uses Vandermonde’s identity on the binomial factor in the closed form (Tankosič, 6 Aug 2025).

The exponential generating function is

$$\genfrac\lvert\rvert{0pt}{}{n}{k}=(-1)^k (n+k)^{\underline n}\,P_n^k(1,1,\dots),$$8

Equivalently,

$$\genfrac\lvert\rvert{0pt}{}{n}{k}=(-1)^k (n+k)^{\underline n}\,P_n^k(1,1,\dots),$$9

This places the Ward-Lah numbers among factorially normalized lower-triangular arrays whose fixed-$n\ge k\ge 1$0 generating functions are rational after reindexing (Tankosič, 6 Aug 2025).

4. Special values and higher-order recurrences

Several special values follow immediately from the explicit formula: $n\ge k\ge 1$1 These entries show the rapid factorial growth along both the first column and the diagonal (Tankosič, 6 Aug 2025).

Using Sister Celine’s general algorithm, the paper derives an order-$n\ge k\ge 1$2 recurrence in the $n\ge k\ge 1$3-direction: $n\ge k\ge 1$4 The role of this recurrence is not to replace the triangular recurrences but to exhibit the holonomic structure of the triangle. In the source, this higher-order relation is grouped with analogous recurrences for the varied and binomial Ward-type arrays (Tankosič, 6 Aug 2025).

A common misconception is to treat Ward-Lah numbers as merely a renamed classical Lah triangle. The formulas above exclude that identification. The binomial factor $n\ge k\ge 1$5 is shared with Lah numbers, but the factorial term is different, and the resulting recurrences, generating function, and special values differ correspondingly.

5. Varied and binomial Ward-Lah numbers

The same paper introduces two further Lah-type Ward families: varied Ward-Lah numbers and binomial Ward-Lah numbers. Their definitions and closed forms can be organized as follows (Tankosič, 6 Aug 2025).

Family	Definition	Explicit formula
Ward-Lah	$n\ge k\ge 1$6	$n\ge k\ge 1$7
Varied Ward-Lah	$n\ge k\ge 1$8	$n\ge k\ge 1$9
Binomial Ward-Lah	$$\genfrac\lvert\rvert{0pt}{}{n}{k}=\frac{(n+k)!}{k!}\binom{n-1}{k-1}.$$0	$$\genfrac\lvert\rvert{0pt}{}{n}{k}=\frac{(n+k)!}{k!}\binom{n-1}{k-1}.$$1

The varied Ward-Lah numbers satisfy

$$\genfrac\lvert\rvert{0pt}{}{n}{k}=\frac{(n+k)!}{k!}\binom{n-1}{k-1}.$$2

and are described through a variation factor involving $$\genfrac\lvert\rvert{0pt}{}{n}{k}=\frac{(n+k)!}{k!}\binom{n-1}{k-1}.$$3. Their triangular recurrence is

$$\genfrac\lvert\rvert{0pt}{}{n}{k}=\frac{(n+k)!}{k!}\binom{n-1}{k-1}.$$4

with exponential generating function

$$\genfrac\lvert\rvert{0pt}{}{n}{k}=\frac{(n+k)!}{k!}\binom{n-1}{k-1}.$$5

The paper notes that this varied Ward-Lah sequence was not yet in the OEIS (Tankosič, 6 Aug 2025).

The binomial Ward-Lah numbers satisfy

$$\genfrac\lvert\rvert{0pt}{}{n}{k}=\frac{(n+k)!}{k!}\binom{n-1}{k-1}.$$6

with boundary conditions

$$\genfrac\lvert\rvert{0pt}{}{n}{k}=\frac{(n+k)!}{k!}\binom{n-1}{k-1}.$$7

Their triangular recurrence is

$$\genfrac\lvert\rvert{0pt}{}{n}{k}=\frac{(n+k)!}{k!}\binom{n-1}{k-1}.$$8

and Sister Celine’s general algorithm yields an order-$$\genfrac\lvert\rvert{0pt}{}{n}{k}=\frac{(n+k)!}{k!}\binom{n-1}{k-1}.$$9 recurrence for this triangle (Tankosič, 6 Aug 2025).

The paper also records a direct identity linking classical Lah numbers to varied Ward-Lah numbers: $n\ge k\ge 1$0 This places the varied Ward-Lah array explicitly inside the classical Lah-number calculus (Tankosič, 6 Aug 2025).

6. Position within the broader Lah-number literature

Ward-Lah numbers belong to a wider family of Lah-type generalizations, but they are not interchangeable with other generalized Lah arrays. The literature summarized in the cited sources includes $n\ge k\ge 1$1-Lah numbers arising from normal ordering in the Weyl algebra (Eu et al., 2017), multiple $n\ge k\ge 1$2-Lah numbers defined as connection coefficients between multiple rising and falling factorial systems (Coskun, 2012), $n\ge k\ge 1$3-Whitney-Lah numbers in the $n\ge k\ge 1$4-Whitney/Dowling framework (Corcino et al., 2020), and $n\ge k\ge 1$5- and $n\ge k\ge 1$6-Lah numbers and distributions with probabilistic interpretations via random compositions, recursive trees, and related structures (Iksanov et al., 2024). In each case, the common theme is a Lah-type connection-coefficient role, but the defining parameters and ambient algebraic structures differ.

The relation to Ward numbers also has to be stated carefully. The paper on Ward numbers and increasing Schröder trees proves that Ward numbers $n\ge k\ge 1$7 satisfy Ward’s recurrence

$n\ge k\ge 1$8

and develops weighted Ward numbers $n\ge k\ge 1$9. In the specialization $$\left\uparrow \begin{matrix} n \ k \end{matrix}\right\downarrow = (n+k-1)\left( \left\uparrow \begin{matrix} n-1 \ k \end{matrix}\right\downarrow + \left\uparrow \begin{matrix} n-1 \ k-1 \end{matrix}\right\downarrow \right),$$0, it derives

$$\left\uparrow \begin{matrix} n \ k \end{matrix}\right\downarrow = (n+k-1)\left( \left\uparrow \begin{matrix} n-1 \ k \end{matrix}\right\downarrow + \left\uparrow \begin{matrix} n-1 \ k-1 \end{matrix}\right\downarrow \right),$$1

where $$\left\uparrow \begin{matrix} n \ k \end{matrix}\right\downarrow = (n+k-1)\left( \left\uparrow \begin{matrix} n-1 \ k \end{matrix}\right\downarrow + \left\uparrow \begin{matrix} n-1 \ k-1 \end{matrix}\right\downarrow \right),$$2 are unsigned Lah numbers. That paper does not define “Ward-Lah numbers” as a named standalone sequence, but it exhibits a Lah-valued specialization inside weighted Ward theory (Wang et al., 21 Jul 2025).

For classical Lah numbers, the Lah triangular matrix $$\left\uparrow \begin{matrix} n \ k \end{matrix}\right\downarrow = (n+k-1)\left( \left\uparrow \begin{matrix} n-1 \ k \end{matrix}\right\downarrow + \left\uparrow \begin{matrix} n-1 \ k-1 \end{matrix}\right\downarrow \right),$$3 is lower triangular, totally non-negative, and therefore variation-decreasing by Motzkin’s theorem. These properties are obtained from a planar-network interpretation and Lindström’s lemma (Martinjak et al., 2017). A plausible implication is that analogous matrix-positivity questions may be asked for Ward-Lah-type matrices as well, but no such theorem is stated in the Ward-Lah paper itself.

Accordingly, Ward-Lah numbers are best understood not as a universal replacement for Lah numbers, nor as a mere reformulation of Ward numbers, but as one specific Ward-type factorial-binomial deformation: $$\left\uparrow \begin{matrix} n \ k \end{matrix}\right\downarrow = (n+k-1)\left( \left\uparrow \begin{matrix} n-1 \ k \end{matrix}\right\downarrow + \left\uparrow \begin{matrix} n-1 \ k-1 \end{matrix}\right\downarrow \right),$$4 together with its varied and binomial companions, its Lah-transform identity, its recurrences, and its generating functions (Tankosič, 6 Aug 2025).