A298804 - OEIS

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

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OFFSET

1,4

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)

LINKS

Table of n, a(n) for n=1..28.

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

EXAMPLE

Triangle begins:

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