

A298804


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


4



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



EXAMPLE

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
...


CROSSREFS



KEYWORD



AUTHOR



STATUS

approved



