OFFSET
0,8
COMMENTS
The leaders of weakly increasing runs in a sequence are obtained by splitting it into maximal weakly increasing subsequences and taking the first term of each.
LINKS
EXAMPLE
Triangle begins:
1
0 1
0 1 1
0 2 0 2
0 3 2 1 2
0 5 4 3 1 3
0 7 10 7 3 1 4
0 11 19 14 9 4 2 5
0 15 39 27 22 10 7 2 6
0 22 69 59 48 24 15 8 3 8
0 30 125 117 104 56 38 19 10 3 10
0 42 211 241 215 132 80 49 25 12 5 12
0 56 354 473 445 296 186 109 61 31 17 5 15
0 77 571 917 896 665 409 258 139 78 41 20 7 18
Row n = 6 counts the following compositions:
. (15) (24) (33) (312) (411) (6)
(114) (141) (231) (3111) (51)
(123) (1311) (213) (2121) (42)
(1113) (1131) (132) (321)
(1122) (222) (2211)
(11112) (1221) (2112)
(111111) (1212) (21111)
(12111)
(11211)
(11121)
MATHEMATICA
Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n], Total[First/@Split[#, LessEqual]]==k&]], {n, 0, 15}, {k, 0, n}]
CROSSREFS
Last column n = k is A000009.
Second column k = 2 is A000041.
Row-sums are A011782.
For length instead of sum we have A238343.
Types of runs (instead of weakly increasing):
- For leaders of constant runs we have A373949.
- For leaders of anti-runs we have A374521.
- For leaders of strictly increasing runs we have A374700.
- For leaders of weakly decreasing runs we have A374748.
- For leaders of strictly decreasing runs we have A374766.
Types of run-leaders:
- For strictly decreasing leaders we appear to have A188920.
- For weakly decreasing leaders we appear to have A189076.
- For identical leaders we have A374631.
- For strictly increasing leaders we have A374634.
- For weakly increasing leaders we have A374635.
A003242 counts anti-run compositions.
A335456 counts patterns matched by compositions.
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Jul 23 2024
STATUS
approved