In this website are resources on counting set partitions and related structures.
It is not a complete reference. These are just some of the things I learned about when studying this topic. I made this website because it would've been of great help to me.
There might be mistakes, double check with the given sources when in doubt.
If there is something you feel should be added, removed or changed, contact me at jpbotelho.costa (at) gmail.com
With no restrictions (Bell Number)
With at least $k$ elements per block -> associated Bell number
With at most $k$ elements per block -> restricted Bell number
Such that the first $k$ elements are in separate blocks -> r-Bell number
Such that elements with pairwise distance $<d$ are in separate blocks (reduced Bell number)
Such that the blocks are ordered but the elements in them aren't -> ordered Bell / Fubini numbers
Into non-crossing partitions -> Catalan number
With block lengths given by the $k$th integer partition of the number of elements -> A036040
Into blocks of equal size -> A038041
Into blocks of different sizes -> A007837
Such that the smallest block has size $k$ -> A182930
...and no restrictions (Stirling number of the second kind)
Such that the blocks aren't ordered but the elements in them are -> Lah numbers
[1] https://oeis.org/wiki/User:Peter_Luschny/SetPartitions
Created on 10/05/2024
Last updated on 14/05/2024