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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.

%I A386891 #9 Aug 08 2025 09:10:55
%S A386891 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,
%T A386891 10,0,271,216,24,1,0,524,447,51,2,0,1031,895,117,5,0,2023,1813,250,10,
%U A386891 0,3998,3630,544,20,0,7878,7344,1115,46,1,0,15601,14738,2330,97,2
%N 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.
%C A386891 Here the set of parts of a composition is the set of all parts appearing in the composition.
%H A386891 John Tyler Rascoe, <a href="/A386891/a386891.py.txt">Python code</a>.
%e A386891 Triangle begins:
%e A386891     k=0    1    2   3  4
%e A386891  n=0  1,
%e A386891  n=1  0,   1,
%e A386891  n=2  0,   2,
%e A386891  n=3  0,   3,   1,
%e A386891  n=4  0,   6,   2,
%e A386891  n=5  0,  11,   5,
%e A386891  n=6  0,  21,  10,  1,
%e A386891  n=7  0,  39,  23,  2,
%e A386891  n=8  0,  74,  49,  5,
%e A386891  n=9  0, 139, 107, 10,
%e A386891  n=10 0, 271, 216, 24, 1,
%e A386891 ...
%e A386891 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.
%e A386891 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.
%o A386891 (Python) # see links
%Y A386891 Cf. A002024 (row lengths), A011782 (row sums).
%Y A386891 Cf. A218796, A333213, A374629, A385604.
%K A386891 nonn,tabf
%O A386891 0,5
%A A386891 _John Tyler Rascoe_, Aug 06 2025