OFFSET
0,10
COMMENTS
The weakly decreasing run-leaders of a sequence are obtained by splitting it into maximal weakly decreasing subsequences and taking the first term of each.
LINKS
EXAMPLE
Triangle begins:
1
0 1
0 1 1
0 1 1 2
0 1 2 3 2
0 1 2 6 4 3
0 1 3 9 8 7 4
0 1 3 13 15 16 11 5
0 1 4 17 24 32 28 16 6
0 1 4 23 36 58 58 44 24 8
0 1 5 28 52 96 115 100 71 34 10
0 1 5 35 72 151 203 211 176 109 49 12
Row n = 6 counts the following compositions:
. (111111) (222) (33) (42) (51) (6)
(2211) (321) (411) (141) (15)
(21111) (3111) (132) (114) (24)
(1221) (1311) (312) (123)
(1122) (1131) (231)
(12111) (1113) (213)
(11211) (2121) (1212)
(11121) (2112)
(11112)
MATHEMATICA
Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n], Total[First/@Split[#, GreaterEqual]]==k&]], {n, 0, 15}, {k, 0, n}]
CROSSREFS
Column n = k is A000009.
Column k = 2 is A004526.
Row-sums are A011782.
For length instead of sum we have A238343.
Column k = 3 is A374702.
The center n = 2k is A374703.
Types of runs (instead of weakly decreasing):
- For leaders of constant runs we have A373949.
- For leaders of anti-runs we have A374521.
- For leaders of weakly increasing runs we have A374637.
- For leaders of strictly increasing runs we have A374700.
- For leaders of strictly decreasing runs we have A374766.
Types of run-leaders:
- For weakly increasing leaders we appear to have A188900.
- For strictly decreasing leaders we have A374746.
- For weakly decreasing leaders we have A374747.
A003242 counts anti-run compositions.
A335456 counts patterns matched by compositions.
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Jul 26 2024
STATUS
approved