A386891 Irregular triangle read by rows: T(n,k) is the number of compositions of n such that the maximal cardinality of C is k, where C is a subset of the set of parts such that all elements in C appear in weakly increasing order within the composition.

1, 0, 1, 0, 2, 0, 3, 1, 0, 6, 2, 0, 11, 5, 0, 21, 10, 1, 0, 39, 23, 2, 0, 74, 49, 5, 0, 139, 107, 10, 0, 271, 216, 24, 1, 0, 524, 447, 51, 2, 0, 1031, 895, 117, 5, 0, 2023, 1813, 250, 10, 0, 3998, 3630, 544, 20, 0, 7878, 7344, 1115, 46, 1, 0, 15601, 14738, 2330, 97, 2

Offset: 0

Author

John Tyler Rascoe, Aug 06 2025

Keywords

• nonn
• tabf

Comments

Here the set of parts of a composition is the set of all parts appearing in the composition.

Examples

			Triangle begins:
    k=0    1    2   3  4
 n=0  1,
 n=1  0,   1,
 n=2  0,   2,
 n=3  0,   3,   1,
 n=4  0,   6,   2,
 n=5  0,  11,   5,
 n=6  0,  21,  10,  1,
 n=7  0,  39,  23,  2,
 n=8  0,  74,  49,  5,
 n=9  0, 139, 107, 10,
 n=10 0, 271, 216, 24, 1,
...
The composition of n = 3 (2,1) with set of parts {1,2} has maximal subsets {1} and {2} both with all parts appearing in weakly increasing order, so (2,1) is counted under T(3,1) = 3.
The composition of n = 15 (3,1,1,2,3,5) with set of parts {1,2,3,5} has the maximal subset {1,2,5}, so (3,1,1,2,3,5) is counted under T(15,3) = 1115.
		

Links

• John Tyler Rascoe, Python code.

Crossrefs

Cf. A002024 (row lengths), A011782 (row sums).

Cf. A218796, A333213, A374629, A385604.

Programs

Python

# see links