Analogs of binomial coefficients

There are several numbers that are analogous to binomial coefficients and, at least in Donald Knuth’s notation, are written in a style analogous to binomial coefficients. And just as binomial coefficients can be arranged into Pascal’s triangle, these numbers can be arranged into similar triangles.

In Pascal’s triangle, each entry is the sum of the two above it. Specifically,

$\binom{n}{k} = \binom{n-1}{k} + \binom{n-1}{k-1}$

The q-binomial coefficients satisfy two similar identities.

$\begin{align*} \binom{n}{k}_q &= q^k \binom{n-1}{k}_q + \binom{n-1}{k-1}_q \\ &= \binom{n-1}{k}_q + q^{n-k}\binom{n-1}{k-1}_q \end{align*}$

Here are the analogous theorems for Stirling numbers of the first

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

and second

$\left\{ \begin{matrix} n \\ k \end{matrix} \right\} = k \left\{ \begin{matrix} n-1 \\ k \end{matrix} \right\} + \left\{ \begin{matrix} n-1 \\ k-1 \end{matrix} \right\}$

kinds.

And finally, here is the corresponding theorem for Eulerian numbers.

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

2 thoughts on “Analogs of binomial coefficients”

Adi Ofer

25 July 2025 at 00:04

Regarding the display of the last formula: the dimensions of the displayed formula are those of the svg file. Something gets messed up between the css rules (height, width, max-width, aspect ratio, and perhaps others) and the settings within the svg files (height, width, viewBox) which causes the displayed dimensions of all the formulas to be those of their respective svg files, and these have varying heights. I’m not exactly sure why this happens (it would require a deeper dive).

If you can change the dimensions in the svg files it should solve the issue. Alternatively one way to work around the issue is to scale the svg files from the html code, e.g. use

(or any other scaling factor you deem appropriate) instead of

(note that there’s no need to have the height and width properties here). It’s not a real solution but it works (I tried it).

HTH.

John

25 July 2025 at 14:25

Thanks.

2 thoughts on “Analogs of binomial coefficients”

Leave a Reply