Set partitions

In this website are resources on counting set partitions and related structures.

How can I calculate these numbers?

Bell number - no restriction
Associated Bell number - $\geq m$ elements per block
Restricted Bell number - $\leq m$ elements per block
r-Bell number - the first $r$ elements are in separate blocks
(S, r)-Bell number - the first $r$ elements are in separate blocks, block cardinality in some set $S$

Stirling number of the 2nd kind - $k$ blocks
Associated Stirling number 2nd kind - $k$ blocks and $\geq m$ elements per block
Restricted Stirling number 2nd kind - $k$ blocks and $\leq m$ elements per block
r-Stirling number 2nd kind - $k$ blocks, the first $r$ elements are in separate blocks
Restricted r-Stirling number 2nd kind - $k$ blocks, the first $r$ elements are in separate blocks, $\leq m$ elements per block
Associated r-Stirling number 2nd kind - $k$ blocks, the first $r$ elements are in separate blocks, $\geq m$ elements per block
(S, r)-Stirling number 2nd kind - $k$ blocks, the first $r$ elements are in separate blocks, block cardinality in some set $S$
Reduced Stirling number 2nd kind - $k$ blocks, elements with pairwise distance $< d$ are in separate blocks

Lah-Bell number - no restrictions
Lah number - $k$ lists
r-Lah number - $k$ lists, first $r$ elements in different lists
r-Lah-Bell number - first $r$ elements in different lists
Restricted r-Lah number - first $r$ elements in different lists, at most $m$ items per list
(S, r)-Lah-Bell number - list cardinality contained in some set $S$ , first $r$ elements in different lists
(S, r)-Lah number - $k$ lists, list cardinality contained in some set $S$ , first $r$ elements in different lists

Fubini number - no restrictions
(S, r)-Fubini number - block cardinality contained in some set $S$ , first $r$ elements in different blocks

Doubly ordered (S, r)-partitions - list cardinality in some set $S$ , first $r$ elements in different lists

João Costa

jpbotelho.costa at gmail dot com