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

0,5

Comments

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

Links

Table of n, a(n) for n=0..66.

John Tyler Rascoe, Python code.

Example

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.

Prog

(Python) # see links

Crossrefs

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

Cf. A218796, A333213, A374629, A385604.

Sequence in context: A209599 A382312 A238347 * A263322 A353496 A170942

Adjacent sequences: A386888 A386889 A386890 * A386892 A386893 A386894

Keyword

nonn,tabf

Author

John Tyler Rascoe, Aug 06 2025

Status

approved