♦ Lah Numbers

♦ As a Rectangle

♦ Interpretations

Posets: Partially ordered sets on n elements that consist entirely of k chains (nonempty, linearly ordered subsets).

Set partitions: Number of ways to split {1,..,n} into an ordered collection of n+1-k nonempty sets that are noncrossing.

Dyck paths: Dyck n-paths with n+1-k peaks labeled 1,2,..n+1-k in some order.

♦ Generators

Recurrence

\begin{equation}T_{n,\,n} = 1, \ T_{n,\,k} = 0 \ (k \lt 0), \\ T_{n,\,k} \ = \ T_{n-1,\,k-1} + (n+k-1)T_{n-1,\,k}\end{equation}

Generating function of triangle row polynomials

\begin{equation}L_n(x) \ = \ \sum_{0 \le k \le n} T_{n,k}\, x^k\end{equation}

\begin{equation}L_n(x) \ = \ n!\, [t^n]\, \exp\left(\frac{tx}{1-t}\right)\end{equation}

Hypergeometric series ₁F ₁.

\begin{equation}L_n(x) \ = \ n! \,x\, F\left(\genfrac{}{\vert}{0pt}{}{-n+1}{2} -x \right) \quad (n \ge 1)\end{equation}

Kummer U.

\begin{equation}L_n(x) \ = \ (-1)^{n-1} \,x\, U_{1-n,2}(-x)\end{equation}

L⁽¹⁾_n generalized Laguerre polynomials.

\begin{equation}L_n(x) \ = \ (n-1)! \,x\, L_{n-1}^{(1)}(-x) \quad (n \ge 1)\end{equation}

Reflected Lah polynomials, Hypergeometric series ₂F ₀.

\begin{equation}x^n \, L_n(1/x) \ = \ F\left(\genfrac{}{\vert}{0pt}{}{-n+1,\,-n}{-} {x} \right) \end{equation}

Generating polynomial of rectangle rows, Stirling cycle number, A254881.

\begin{equation}S_n(x) \ = \ \delta_{n,0} + \sum_{k=0}^{2n} x^k \sum_{j=0}^{k-1} \genfrac{ [ }{ ] }{0pt}{}{n+1}{j+1} \genfrac{ [ }{ ] }{0pt}{}{n}{k-j}\end{equation}

\begin{equation}T_{n+k,\, k} \ = \ S_n(k) / n!\end{equation}

Generating rational function of rectangle rows, A253284 .

\begin{equation}G_n(x) \ = \ -\sum_{k=0}^n \frac{(n+1)!}{k+1} \binom{n+k}{n} \binom{n}{k} (x-1)^{-(n+k+1)}\end{equation}

\begin{equation}T_{n+k,\, k} \ = \ [x^k] \, G_n(x)\end{equation}

Generating exponential function of rectangle columns, A253283.

\begin{equation}R_n(x) \ = \ \frac{d^n}{d x^n} \left( \frac{1}{n!} \left( \frac{x}{1-x} \right)^n \right)\end{equation}

\begin{equation}T_{n+k,\, n} \ = \ k! \, [x^k] R_n(x) \quad (n \ge 1)\end{equation}

Exponential Riordan array.

\begin{equation}T_{n,\, k} \ = \ \frac{n!}{k!} \left[ 1 \mid \frac{x}{1-x} \right]_{n,\ k}\end{equation}

♦ Integer Formulas

For all integers n, k: Stirling cycle number, Stirling subset number.

\begin{equation}T_{n,\,k} \ = \ \sum \limits_{j\,=\,k}^{n} \genfrac[]{0pt}{}{n}{j} \genfrac\{\}{0pt}{}{j}{k}\end{equation}

Triangle for 0 ≤ k ≤ n:

\begin{equation}T_{n,\,k} \ = \ (n-k)! \binom{n}{n-k} \binom{n-1}{n-k}\end{equation}

A103371.

\begin{equation}\frac{T_{n,\,k}}{(n-k)!} \ = \ \binom{n}{n-k} \binom{n-1}{n-k}\end{equation}

Little Narayana numbers N(n,k), A090181.

\begin{equation}T_{n,\,k} \ = \ (n-k+1)! \, N_{n,\,k}\end{equation}

Triangle, T_0,0 = 1 and for 0 ≤ k ≤ n and n > 0:

\begin{equation}T_{n,\,k} \ = \ \frac{n\,k}{(n-k)!} \left((n-1)^\underline{n-k-1}\right)^2 \quad (n \ge 1)\end{equation}

Rows of rectangle

\begin{equation}T_{n+k,\,k} \ = \ \frac{(n+k)!}{k!} \binom{n+k-1}{k-1}\end{equation}

Columns of rectangle

\begin{equation}T_{n+k,\,n} \ = \ \frac{(n+k)!}{n!} \binom{n+k-1}{n-1} \quad (n \ge 1)\end{equation}

Diagonal of rectangle, C_n Catalan number.

\begin{equation}T_{2n,\,n} \ = \ (n+1)! \binom{2n-1}{n} \, C_{n}\end{equation}

Diagonal of rectangle, Γ function.

\begin{equation}T_{2n,\,n} \ = \ \frac{2}{n} \frac{\Gamma(2n)^2}{\Gamma(n)^3} \quad (n \ge 1)\end{equation}

♦ Transforms

Transform: $\ \mathbb{T}_n(A_k) = \sum_{k=0}^{n}T_{n,k}\,A_k.$

Inverse transform: $\ \mathbb{T}_n^{*}(A_k) = \sum_{k=0}^{n}(-1)^{n-k}\,T_{n,k}\,A_k.$

Row sums, Stirling cycle number, B_k Bell number.

\begin{equation}\mathbb{T}_n(1) \ = \ \sum \limits_{k=0}^{n} \genfrac[]{0pt}{}{n}{k} B_{k}\end{equation}

Row sums, Kummer U.

\begin{equation}\mathbb{T}_n(1) \ = \ (-1)^{n-1} U_{1-n,2}(-1)\end{equation}

Alternating row sums, Hypergeometric series ₂F ₀.

\begin{equation}\mathbb{T}_n^{*}(1) \ = \ F\left(\genfrac{}{\vert}{0pt}{}{-n+1,\,-n}{-} -1\right)\end{equation}

A052852, Hypergeometric series ₁F ₁.

\begin{equation}\mathbb{T}_n(k) \ = \ \frac{(n+1)!}{2} F\left(\genfrac{}{\vert}{0pt}{}{-n+1}{3} -1\right) \quad (n \ge 1)\end{equation}

A103194, Hypergeometric series ₁F ₁.

\begin{equation}\mathbb{T}_n(k^{2}) \ = \ n\, n! \, F\left(\genfrac{}{\vert}{0pt}{}{-n+1}{2} -1\right)\end{equation}

A256467, Hypergeometric series ₂F ₂.

\begin{equation}\mathbb{T}_n^{*}(k^{2}) \ = \ (-1)^{n+1}\, n! \, F\left(\genfrac{}{\vert}{0pt}{}{-n+1, 2}{1, 1} 1 \right) \quad (n \ge 1)\end{equation}

A002866.

\begin{equation}\mathbb{T}_n(k!) \ = \ n! \, 2^{n-1} \quad (n \ge 1)\end{equation}

L_n Laguerre polynomials, A002720.

\begin{equation}\mathbb{T}_n(k+1) \ = \ n! \, L_{n}(-1)\end{equation}

L_n Laguerre polynomials, A202410.

\begin{equation}\mathbb{T}_n^{*}(k+1) \ = \ (-1)^n \, n! \, (L_{n}(1) - 2\,L_{n-1}(1) ) \quad (n\ge 1)\end{equation}

Connecting constants falling factorials, rising factorials.

\begin{equation}\mathbb{T}_n(x^{\underline{k}}) \ = \ x^{\overline{n}}\end{equation}

P_{n, k}(s) Partition transform of sequence s.

\begin{equation}T_{n,\,k} \ = \ (-1)^{k} \frac{n!}{k!} \operatorname{P}_{n,k}( 1,1,1, \ldots)\end{equation}

B_{n, k}(s) Bell transform of sequence s.

\begin{equation}T_{n,\,k} \ = \ \operatorname{B}_{n,k}(m! - 0^m; m \ge 0)\end{equation}

Matrix inverse A111596.

\begin{equation}\operatorname{Mat}^{-1}[T_{n,k}] \ = \ \operatorname{Mat}[(-1)^{n-k}T_{n,k}]\end{equation}

Exponential Riordan array, S_{n, k} Stirling cycle number, A131222.

\begin{equation}\operatorname{Mat}[T_{n,k}] \cdot \operatorname{Mat}[S_{n,k}] \ = \ \frac{n!}{k!} \left[ 1 \mid \log\left( \frac{1-x}{1-2x} \right) \right]_{n,\, k}\end{equation}

Exponential Riordan array, S_{n, k} Stirling set number, A256893.

\begin{equation}\operatorname{Mat}[S_{n,k}] \cdot \operatorname{Mat}[T_{n,k}] \ = \ \frac{n!}{k!} \left[ 1 \mid \frac{1}{2 - \exp(x)} \right]_{n,\, k}\end{equation}

♦ Extensions

Natural extension to n, k in Z, compare formula 15.

\begin{equation}\overline{T}(n,k) = \begin{cases} T(n,k) & \text{if } n \geq 0, \, 0 \leq k \leq n; \\ T(-k,-n) & \text{if } n < 0, \, k < 0; \\ 0 & \text{otherwise.} \end{cases}\end{equation}

Lah numbers of order m, parametrization with the same boundary values as in (1).

\begin{equation}T_{m}(n,k) = T_{m}(n-1,k-1) + (k^m + (n-1)^m)\ T_{m}(n-1,k)\end{equation}

Half integer values, rising factorials.

\begin{equation}T\left(n+\frac12, \, k+\frac12\right) \ = \ \frac{(n+1)^{\overline{n}}} {(k+1)^{\overline{k}}} \frac{(n+2)^{\overline{n+1}}} {(k+2)^{\overline{k+1}}} \frac{16^{k-n}}{(n-k)!} \end{equation}

Complex function interpolating the triangle (the limit is understood where appropriate).

\begin{equation}T(z, w) \ = \ \frac{z\,w }{\Gamma(z-w+1)}\left(\frac{\Gamma(z)}{\Gamma(w+1)}\right)^2\end{equation}

♦ Polynomial values

$V_{n,k} = L_{n}(k)$, the Lah polynomials $L_{n}(x)$ are defined in (2) and (3), A253286.