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The Lah numbers count ordered-block set partitions and expand the rising factorial in the falling basis

Statement

Let $L (0, 0) : = 1$ , let $L (n, 0) : = 0$ for $n \geq 1$ , and for $1 \leq k \leq n$ define

$L (n, k) : = \frac{n!}{k!} (\binom{n - 1}{k - 1}) .$

Then for every $n, m \in N$ ,

$m^{\overline{n}} = \sum_{k = 0}^{n} L (n, k) m^{\underline{k}} .$

Moreover, $L (n, k)$ counts partitions of $[n]$ into exactly $k$ nonempty blocks, each equipped with a linear order.

Facts & Assumptions

Given: The rising and falling factorials of The rising factorial and The factorial $n!$ and the falling factorial $n^{\underline{k}}$ , defined by recursion in $N$ .

Proof

technique · direct

1.1

Fix $n \geq 1$ and $k \in {1, \dots, n}$ . Take a permutation of $[n]$ , written as a word of length $n$ , and choose $k - 1$ of the $n - 1$ gaps between consecutive letters. Cutting the word at those gaps produces an ordered list of $k$ nonempty ordered blocks. This gives $n! (\binom{n - 1}{k - 1})$ ordered lists of ordered blocks.

given

2.1

Forgetting the left-to-right order of the $k$ blocks divides by $k!$ , because every unordered family of $k$ internally ordered blocks has exactly $k!$ linear orders of its blocks. Hence $L (n, k)$ counts partitions of $[n]$ into $k$ nonempty linearly ordered blocks.

step 1.1algebra

3.1

For $m \in N$ , consider $m$ distinguishable boxes arranged from left to right. Building an ordered list inside each box by inserting the elements $1, 2, \dots, n$ one after another gives $m (m + 1) \dots (m + n - 1) = m^{\overline{n}}$ possibilities. Grouping the outcomes by the number $k$ of nonempty boxes, one first chooses the underlying partition of $[n]$ into $k$ internally ordered blocks, counted by $L (n, k)$ from step 2.1, and then chooses the $k$ occupied boxes in order, which gives $m^{\underline{k}}$ possibilities. Summing over $k$ yields $m^{\overline{n}} = \sum_{k = 0}^{n} L (n, k) m^{\underline{k}}$ .

step 2.1given

4.1

Steps 2.1 and 3.1 prove the counting interpretation and the falling-factorial expansion on natural arguments, together with the defining value $L (0, 0) = 1$ .

step 2.1step 3.1∎

Depends on

Used by

The Lah number L(4,2) Example

Dependency tree · two levels

16 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Kenneth P. Bogart, Combinatorics Through Guided Discovery, §3.1.5 (standard reference, not scraped)