OFFSET
0,10
EXAMPLE
Triangle begins:
1
0 1
0 1 1
0 1 0 3
0 1 1 2 4
0 1 0 4 4 7
0 1 1 5 6 5 14
0 1 0 6 10 10 14 23
0 1 1 6 14 12 29 26 39
0 1 0 9 16 19 40 54 46 71
0 1 1 8 22 22 64 82 96 92 124
0 1 0 10 26 30 82 137 144 204 176 214
0 1 1 11 32 31 121 186 240 331 393 323 378
Row n = 6 counts the following compositions:
. (111111) (222) (33) (3111) (411) (6)
(2211) (1113) (114) (51)
(1122) (1221) (1311) (15)
(21111) (12111) (1131) (42)
(11112) (11211) (2112) (24)
(11121) (141)
(321)
(312)
(231)
(213)
(132)
(123)
(2121)
(1212)
For example, the composition (1,2,2,1) with compression (1,2,1) is counted under T(6,4).
MATHEMATICA
Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n], Total[First/@Split[#]]==k&]], {n, 0, 10}, {k, 0, n}]
CROSSREFS
Column k = n is A003242 (anti-runs or compressed compositions).
Row-sums are A011782.
Same as A373951 with rows reversed.
Column k = 3 is A373952.
A114901 counts compositions with no isolated parts.
A240085 counts compositions with no unique parts.
A333755 counts compositions by compressed length.
A373948 represents the run-compression transformation.
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Jun 28 2024
STATUS
approved