Generalised Lah Numbers: Extensions & Applications

• Generalised Lah numbers are extensions of classical Lah numbers that count partitions of a set into ordered blocks and serve as connection coefficients between rising and falling factorials.
• They encompass distinct constructions such as higher-level numbers with shared block leaders and order‑s numbers defined via higher-level Stirling convolutions, each with precise recurrence relations.
• Applications range from operator algebras and Sheffer arrays to multivariate total-positivity frameworks, offering fresh insights into combinatorial enumeration and algebraic transformations.

Generalised Lah numbers are extensions of the classical Lah numbers $L(n,k)$, which count partitions of $[n]$ into $k$ nonempty linearly ordered blocks and simultaneously act as connection coefficients between rising and falling factorial bases. The subject comprises several non-equivalent but closely related constructions. A particularly sharp distinction is between the combinatorial Lah numbers with higher level and the algebraic Lah numbers of order $s$, introduced as complementary generalisations of the classical array; surrounding literature develops $r$-, $(r,s)$-, restricted, Whitney-type, operator-theoretic, multivariate, and heterogeneous variants (Tankosič, 30 Oct 2025).

1. Classical Lah numbers as the prototype

The classical Lah numbers $L(n,k)$ are defined for $0 \le k \le n$ by the connection-coefficient identities

$$x^{\overline{n}}=\sum_{k=0}^{n} L(n,k)\,x^{\underline{k}}, \qquad x^{\underline{n}}=\sum_{k=0}^{n} (-1)^{n+k}\,L(n,k)\,x^{\overline{k}},$$

where

$$x^{\overline{n}}=x(x+1)\cdots(x+n-1), \qquad x^{\underline{n}}=x(x-1)\cdots(x-n+1), \qquad x^{\overline{0}}=x^{\underline{0}}=1.$$

Combinatorially, $[n]$0 counts partitions of $[n]$1 into $[n]$2 nonempty linearly ordered blocks. Its standard closed form and triangular recurrence are

$[n]$3

$[n]$4

with boundary data

$[n]$5

These formulas define the reference model against which later generalisations are measured (Tankosič, 30 Oct 2025).

The classical array already exhibits the two principal themes that persist in the generalised theory. One theme is combinatorial: ordered blocks, list structure, and refinements by distinguished elements or allowed block sizes. The other is algebraic: transition matrices between factorial-type polynomial bases. Most later constructions preserve one of these themes exactly and modify the other.

2. The 2025 bifurcation: higher-level Lah numbers and Lah numbers of order $[n]$6

The paper "The Lah Numbers with Higher Level and the Lah Numbers of Order $[n]$7" introduces two distinct extensions indexed by $[n]$8 (Tankosič, 30 Oct 2025). The first is combinatorial. The Lah numbers with higher level,

$[n]$9

count ordered $k$0-tuples $k$1 of partitions of $k$2 into $k$3 lists such that the sets of block leaders coincide: $k$4 They satisfy

$k$5

with

$k$6

and vanish for $k$7. Two special cases are

$k$8

The second extension is algebraic. The Lah numbers of order $k$9,

$s$0

are defined by the higher-level Stirling convolution

$s$1

where $s$2 and $s$3 are the higher-level Stirling numbers of the first and second kinds. Their recurrence is

$s$4

with the same triangular boundary pattern. Their distinguished role is as connection coefficients between higher-level rising and falling factorials

$s$5

through

$s$6

The two constructions coincide with the classical Lah numbers when $s$7: $s$8

3. Structural comparison and internal relations

The two 2025 constructions are complementary rather than redundant. The higher-level numbers preserve a direct combinatorial interpretation in terms of shared block leaders, while the order-$s$9 numbers are tuned to the basis transformation

$r$0

The paper explicitly states that the higher-level Lah numbers do not furnish connection identities with higher-level falling and rising factorials, whereas the order-$r$1 Lah numbers are defined precisely to provide those missing identities (Tankosič, 30 Oct 2025).

Aspect	Higher level	Order $r$2
Definition	ordered $r$3-tuples of list partitions with common $r$4	$r$5
Main recurrence term	$r$6	$r$7
Primary role	combinatorial refinement	factorial-basis connection coefficients

A direct theorem identifies the higher-level Lah numbers with a special case of $r$8-Lah numbers: $r$9 This exact specialisation means that all identities for the higher-level array follow from the corresponding formulas for $(r,s)$0-Lah numbers at $(r,s)$1 or $(r,s)$2. The order-$(r,s)$3 numbers, by contrast, admit a matrix interpretation: the infinite matrices of higher-level Stirling numbers of both kinds and Lah numbers of order $(r,s)$4, together with their signed analogues, act as transition matrices between the bases $(r,s)$5, $(r,s)$6, and $(r,s)$7 in $(r,s)$8 (Tankosič, 30 Oct 2025).

The same paper also gives a hierarchy

$(r,s)$9

and the linking identity at $L(n,k)$0,

$L(n,k)$1

These formulas make the contrast quantitative: the combinatorial construction is larger, and the algebraic construction sits between it and the higher-level Stirling arrays.

4. Other major families of generalised Lah numbers

Beyond the higher-level/order-$L(n,k)$2 split, the literature uses the phrase “generalised Lah numbers” for several other extensions. One prominent axis is the $L(n,k)$3- and $L(n,k)$4-theory. The $L(n,k)$5-Lah numbers satisfy

$L(n,k)$6

and admit the convolution

$L(n,k)$7

The more general $L(n,k)$8-Lah distributions are probability laws on $L(n,k)$9 with mass

$0 \le k \le n$0

and are represented both by subtree sizes in a Hoppe tree with split root weight $0 \le k \le n$1 and by sums of Bernoulli counts driven by a Dirichlet-multinomial composition (Iksanov et al., 2024).

A second axis imposes block-size restrictions. For $0 \le k \le n$2, the $0 \le k \le n$3-Lah number $0 \le k \le n$4 counts partitions of $0 \le k \le n$5 into $0 \le k \le n$6 ordered lists whose sizes lie in $0 \le k \le n$7, while $0 \le k \le n$8 additionally requires the first $0 \le k \le n$9 elements to lie in distinct blocks. Their fixed-$$x^{\overline{n}}=\sum_{k=0}^{n} L(n,k)\,x^{\underline{k}}, \qquad x^{\underline{n}}=\sum_{k=0}^{n} (-1)^{n+k}\,L(n,k)\,x^{\overline{k}},$$0 exponential generating functions are

$$x^{\overline{n}}=\sum_{k=0}^{n} L(n,k)\,x^{\underline{k}}, \qquad x^{\underline{n}}=\sum_{k=0}^{n} (-1)^{n+k}\,L(n,k)\,x^{\overline{k}},$$1

$$x^{\overline{n}}=\sum_{k=0}^{n} L(n,k)\,x^{\underline{k}}, \qquad x^{\underline{n}}=\sum_{k=0}^{n} (-1)^{n+k}\,L(n,k)\,x^{\overline{k}},$$2

This framework contains associated, restricted, odd, even, and $$x^{\overline{n}}=\sum_{k=0}^{n} L(n,k)\,x^{\underline{k}}, \qquad x^{\underline{n}}=\sum_{k=0}^{n} (-1)^{n+k}\,L(n,k)\,x^{\overline{k}},$$3-Lah cases, and it is naturally described by exponential Riordan arrays (Bényi et al., 2020). A related restricted-block-size theory writes $$x^{\overline{n}}=\sum_{k=0}^{n} L(n,k)\,x^{\underline{k}}, \qquad x^{\underline{n}}=\sum_{k=0}^{n} (-1)^{n+k}\,L(n,k)\,x^{\overline{k}},$$4 for partitions into lists with all block sizes in $$x^{\overline{n}}=\sum_{k=0}^{n} L(n,k)\,x^{\underline{k}}, \qquad x^{\underline{n}}=\sum_{k=0}^{n} (-1)^{n+k}\,L(n,k)\,x^{\overline{k}},$$5; the inverse matrices $$x^{\overline{n}}=\sum_{k=0}^{n} L(n,k)\,x^{\underline{k}}, \qquad x^{\underline{n}}=\sum_{k=0}^{n} (-1)^{n+k}\,L(n,k)\,x^{\overline{k}},$$6 exist if and only if $$x^{\overline{n}}=\sum_{k=0}^{n} L(n,k)\,x^{\underline{k}}, \qquad x^{\underline{n}}=\sum_{k=0}^{n} (-1)^{n+k}\,L(n,k)\,x^{\overline{k}},$$7, and their entries admit forest interpretations, in general as differences of even and odd ordered phylogenetic forest counts and, under a “no exposed odds” condition, as signed cardinalities of single families of $$x^{\overline{n}}=\sum_{k=0}^{n} L(n,k)\,x^{\underline{k}}, \qquad x^{\underline{n}}=\sum_{k=0}^{n} (-1)^{n+k}\,L(n,k)\,x^{\overline{k}},$$8-good forests (Engbers et al., 2016).

A third axis modifies the Stirling side rather than the block structure. The heterogeneous Stirling numbers

$$x^{\overline{n}}=\sum_{k=0}^{n} L(n,k)\,x^{\underline{k}}, \qquad x^{\underline{n}}=\sum_{k=0}^{n} (-1)^{n+k}\,L(n,k)\,x^{\overline{k}},$$9

interpolate between Stirling and Lah arrays: $$x^{\overline{n}}=x(x+1)\cdots(x+n-1), \qquad x^{\underline{n}}=x(x-1)\cdots(x-n+1), \qquad x^{\overline{0}}=x^{\underline{0}}=1.$$0 and $$x^{\overline{n}}=x(x+1)\cdots(x+n-1), \qquad x^{\underline{n}}=x(x-1)\cdots(x-n+1), \qquad x^{\overline{0}}=x^{\underline{0}}=1.$$1 as $$x^{\overline{n}}=x(x+1)\cdots(x+n-1), \qquad x^{\underline{n}}=x(x-1)\cdots(x-n+1), \qquad x^{\overline{0}}=x^{\underline{0}}=1.$$2. Their recurrence

$$x^{\overline{n}}=x(x+1)\cdots(x+n-1), \qquad x^{\underline{n}}=x(x-1)\cdots(x-n+1), \qquad x^{\overline{0}}=x^{\underline{0}}=1.$$3

specialises at $$x^{\overline{n}}=x(x+1)\cdots(x+n-1), \qquad x^{\underline{n}}=x(x-1)\cdots(x-n+1), \qquad x^{\overline{0}}=x^{\underline{0}}=1.$$4 to the classical Lah recurrence, and the associated heterogeneous Bell polynomials interpolate between Bell and Lah–Bell polynomials (Kim et al., 30 Mar 2025).

A fourth axis introduces extra weights. The translated Whitney-Lah numbers satisfy

$$x^{\overline{n}}=x(x+1)\cdots(x+n-1), \qquad x^{\underline{n}}=x(x-1)\cdots(x-n+1), \qquad x^{\overline{0}}=x^{\underline{0}}=1.$$5

so they are a direct scalar deformation of classical Lah numbers. The same paper develops translated $$x^{\overline{n}}=x(x+1)\cdots(x+n-1), \qquad x^{\underline{n}}=x(x-1)\cdots(x-n+1), \qquad x^{\overline{0}}=x^{\underline{0}}=1.$$6-Whitney-Lah numbers and a corresponding $$x^{\overline{n}}=x(x+1)\cdots(x+n-1), \qquad x^{\underline{n}}=x(x-1)\cdots(x-n+1), \qquad x^{\overline{0}}=x^{\underline{0}}=1.$$7-Guo–Qi identity (Mangontarum, 2020). The two-parameter family $$x^{\overline{n}}=x(x+1)\cdots(x+n-1), \qquad x^{\underline{n}}=x(x-1)\cdots(x-n+1), \qquad x^{\overline{0}}=x^{\underline{0}}=1.$$8 weights $$x^{\overline{n}}=x(x+1)\cdots(x+n-1), \qquad x^{\underline{n}}=x(x-1)\cdots(x-n+1), \qquad x^{\overline{0}}=x^{\underline{0}}=1.$$9-Lah distributions by record-low statistics,

$[n]$00

and specialises simultaneously to $[n]$01-Lah and to both kinds of $[n]$02-Stirling numbers (Shattuck, 2014).

5. Operator-theoretic, Sheffer, and multivariate generalisations

One influential algebraic framework comes from normal ordering in the Weyl algebra $[n]$03. For any word $[n]$04 with $[n]$05 $[n]$06's and $[n]$07 $[n]$08's that starts with $[n]$09, the $[n]$10-Lah numbers $[n]$11 are defined by

$[n]$12

These coefficients admit graph-theoretic interpretations as counts of partitions of a quasi-threshold graph $[n]$13 into disjoint unions of $[n]$14 decreasing forests, rook-theoretic interpretations on Ferrers boards, $[n]$15-analogues in the $[n]$16-deformed Weyl algebra, and a rook factorisation

$[n]$17

At $[n]$18, these generalised coefficients reduce to the classical Lah numbers (Eu et al., 2017).

A second algebraic framework treats Lah numbers as a Sheffer array. For parameters $[n]$19, the generalised factorials

$[n]$20

are connected by generalised Lah numbers $[n]$21: $[n]$22 The corresponding Sheffer pair is

$[n]$23

and the inverse matrix satisfies

$[n]$24

This construction recovers the classical unsigned Lah numbers at $[n]$25 (Lang, 2017).

A third direction is multivariate. The generic Lah polynomials $[n]$26 enumerate unordered forests of increasing ordered trees on $[n]$27 with a weight $[n]$28 for each vertex with $[n]$29 children. If the sequence $[n]$30 is Toeplitz-totally positive, then the lower-triangular matrix $[n]$31 is totally positive and the row-generating polynomials are coefficientwise Hankel-totally positive. Positive-type multivariate Lah polynomials also admit an $[n]$32-branched Stieltjes-type continued fraction (Pétréolle et al., 2019). Closely related is a five-variable Laguerre-digraph array $[n]$33 that generalises rook and Lah polynomials; the Lah regime occurs at $[n]$34, where cycles are forbidden, and the resulting arrays again satisfy total-positivity and Hankel-total-positivity statements under explicit coefficientwise inequalities (Deb et al., 2023).

A fourth direction uses multiple logarithms. The multi-Lah numbers $[n]$35 are defined by

$[n]$36

and reduce to the classical Lah numbers when all indices equal $[n]$37: $[n]$38 They sit beside multi-Stirling and multi-Bernoulli numbers inside a common multiple-polylogarithmic framework (Kim et al., 2023).

6. Computation, examples, and current directions

The newer higher-level/order-$[n]$39 theory comes with explicit recurrences, differential recurrences for row polynomials, and small tables. For the higher-level numbers, the row polynomial

$[n]$40

satisfies a derivative recurrence involving classical Stirling numbers of the second kind. For the order-$[n]$41 numbers, the row polynomial

$[n]$42

obeys

$[n]$43

The paper recommends using these recurrences and the boundary conditions for practical computation, and it records the explicit values

$[n]$44

and

$[n]$45

The combinatorial example

$[n]$46

is obtained by leader-set counting: $[n]$47 for the three possible leader sets $[n]$48, $[n]$49, and $[n]$50 (Tankosič, 30 Oct 2025).

A parallel development concerns shape properties of rows. Triangular arrays satisfying the super-recurrence

$[n]$51

have log-concave rows under explicit sufficient conditions. This includes the generalised Lah recurrence

$[n]$52

as well as the $[n]$53-Lah numbers

$[n]$54

The result confirms Tankosić’s conjecture that rows of $[n]$55-Lah numbers are log-concave in $[n]$56 (Shankar, 17 Aug 2025).

Several open directions are stated explicitly in the literature. For higher-level and order-$[n]$57 Lah numbers, closed-form OGFs, EGFs, and bivariate EGFs are not provided, and deriving them is identified as an interesting direction (Tankosič, 30 Oct 2025). For generic multivariate Lah polynomials, a branched continued fraction for the negative-type specialisation is presented as an open direction, and a simple closed form in the $[n]$58-variables is described as elusive (Pétréolle et al., 2019). For restricted Lah matrices, open problems include characterising those $[n]$59 for which the relevant compositional inverses have alternating coefficients and extending the involutive forest constructions beyond the “no exposed odds” regime (Engbers et al., 2016).

Taken together, these developments show that “generalised Lah numbers” is not a single sequence but a family of frameworks. Some preserve the ordered-block combinatorics and refine it by leaders, restrictions, or distinguished elements; others preserve the connection-coefficient role and modify the underlying factorial bases; still others embed Lah numbers into operator algebras, Riordan arrays, total-positivity theories, or probabilistic models. The modern theory is therefore best understood not as one extension, but as a network of extensions centred on the dual classical role of Lah numbers as both list-partition enumerators and factorial-basis transition coefficients.