Partition related number triangles

Set partitions by number of blocks
All partitions of a set with n elements into k blocks are enumerated by the number triangle called Stirling numbers of the second kind. When some partitions are left out or treated as equivalent the new set of partitions is enumerated by a new number triangle. The number triangles for noncrossing partitions are symmetric.

In the two illustrations above some set partitions, sharing the same colors, are grouped together. They always correspond to the same integer partition, shown in the following illustration:

Lah numbers
Counting not only the partitions (which gives the Stirling2 numbers shown above), but also the possible permutations within the blocks gives the triangle of unsigned Lah numbers (with the row sums ).

Also here one could define a noncrossing version (where T(4;2) would be 32 instead of 36).



Set partitions by type and by number of singletons


Subsequences
The by type triangles have some interesting subsequences which are highlighted by colors in the files linked below. 1 * n + n * 1 stands for the integer partition with one addend n and n addends 1, etc.

N  =  0, 1, 2,  3,  4,   5,   6...  (N)  =  1, 2, 6, 20, 70, 252, 924... (central binomial coefficients)
 * The sequence 1,6,20,70,252,924... appears as All, All ■ and All, Noncrossing ■ :
 * There is one partition of a 2-set that has a 1-block (i.e. a singleton) and another singleton.
 * For n>1 there are (n) partitions of a 2n-set that have an n-block and n singletons.

N  =  0, 1,  2,  3,   4,   5...  (N)  =  1, 3, 10, 35, 126, 462...
 * All, All □ :
 * There are (n+1) partitions of a 2n-set that have two n-blocks.


 * All, All ■ and All, Noncrossing  ■ :
 * There is one partition of a 3-set that has a 1-block (i.e. a singleton) and two other singletons.
 * For n>1 there are (n) partitions of a (2n+1)-set that have an n-block and n+1 singletons.


 * All, All ■ and All, Noncrossing ■ :
 * There are (n) partitions of a (2n+1)-set that have an (n+1)-block and n singletons.


 * All, All □ :
 * There are (n) partitions of a (2n+1)-set that have an (n+1)-block and an n-block.

N+  =  1, 2, 3,  4,  5,  6...  (N+)  =  1, 2, 4, 10, 26, 80...
 * Up to rot, All ■ and Up to rot, Noncrossing ■ :
 * There are (n) necklaces of length 2n with n black and n white beads.


 * Up to rot & ref, All ■ ■  □ and Up to rot & ref, Noncrossing ■  ■ :
 * There are (n) free necklaces (or bracelets) of length 2n+1 with n black and n+1 white beads.