A298804 Triangle T(n,k) (1 <= k <= n) read by rows: A046936 with rows reversed and offset changed to 1.

0, 1, 1, 3, 2, 1, 9, 6, 4, 3, 31, 22, 16, 12, 9, 121, 90, 68, 52, 40, 31, 523, 402, 312, 244, 192, 152, 121

Offset: 1

Author

N. J. A. Sloane, Jan 30 2018, following a suggestion from Don Knuth, Jan 29 2018

Keywords

• nonn
• tabl
• more

Comments

This is another version of Moser's version (A046936) of Aitken's array (A011971).

Although offset 0 is better for A011971 and A046936, for this version offset 1 is more appropriate.

Comments from Don Knuth, Jan 29 2018 (Start):

a(n,k) is the number of set partitions (i.e. equivalence classes) in which (i) 1 is not equivalent to 2, ..., nor k; and (ii) the last part, when parts are ordered by their smallest element, has size 1; (iii) that last part isn't simply "1". (Equivalently, n>1.)

It's not difficult to prove this characterization of a(k,n). For example, if we know that there are 22 partitions of {1,2,3,4,5} with 1 inequivalent to 2, and 6 partitions of {1,2,3,4} with

1 inequivalent to 2, then there are 6 partitions of {1,2,3,4,5} with 1 inequivalent to 2 and 1 equivalent to 3. Hence there are 16 with 1 equivalent to neither 2 nor 3.

The same property, but leaving out conditions (ii) and (iii), characterizes Pierce's triangular array A123346. (End)

Examples

			Triangle begins:
0,
1, 1,
3, 2, 1,
9, 6, 4, 3,
31, 22, 16, 12, 9,
121, 90, 68, 52, 40, 31
523, 402, 312, 244, 192, 152, 121
...
		

Links

• Don Knuth, Email to N. J. A. Sloane, Jan 29 2018

Crossrefs

Cf. A011971, A040027, A046936, A123346.