A171871 Triangle read by rows: Distinct classifications of N elements containing exactly R binary partitions.

1, 0, 1, 0, 0, 1, 0, 0, 1, 2, 0, 0, 0, 3, 3, 0, 0, 0, 3, 17, 6, 0, 0, 0, 1, 36, 74, 11, 0, 0, 0, 1, 60, 573, 358, 23, 0, 0, 0, 0, 56, 2802, 7311, 1631, 47, 0, 0, 0, 0, 50, 10087, 107938, 83170, 7563, 106, 0, 0, 0, 0, 27, 26512, 1186969, 3121840, 866657, 34751, 235, 0, 0, 0, 0, 19

Offset: 0

Author

Robert Munafo, Jan 21 2010

Keywords

• nonn
• tabl

Comments

Significance of triangle suggested by Franklin T. Adams-Watters on Dec 19 2009. Row N has N terms in this sequence. The triangle starts:

1;

0, 1;

0, 0, 1;

0, 0, 1, 2;

0, 0, 0, 3, 3;

0, 0, 0, 0, 3, 17, 6;

0, 0, 0, 0, 1, 36, 74, 11;

Value is A000055(N) when R=N-1 (last term in each row). (Conjectured by Robert Munafo Dec 28 2009, then proved by Andrew Weimholt and Franklin T. Adams-Watters on Dec 29 2009)

Value is 1 when N=2^R.

Value is 1 when N=(2^R)-1.

Value is R when R>2 and N=(2^R)-2.

Value is A034198(R) when R>2 and N=(2^R)-3.

Conjecture: In general, in each column, the last 2^(R-1) values are the same as the first 2^(N-1) values from the corresponding row of A039754. - Robert Munafo, Dec 30 2009

Value is 0 for all (N,R) for which N is greater than 2^R.

Each term A(N,R) can be computed most efficiently by first enumerating all classifications in A(N-1,R) plus those in A(N-1,R-1), and then adding an additional type and/or partition to each.

Links

• R. Munafo, Classifications of N Elements

Crossrefs

Cf. Row sums are A005646, column sums are A171832.

Cf. A039754.

Last term in each row is A000055(N).

Same triangle read by columns is A171872.