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A374637
Triangle read by rows where T(n,k) is the number of integer compositions of n whose leaders of weakly increasing runs sum to k.
22
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
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.
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.
The corresponding rank statistic is A374630, row-sums of A374629.
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 distinct leaders we have A374632, ranks A374768.
- For strictly increasing leaders we have A374634.
- For weakly increasing leaders we have A374635.
A003242 counts anti-run compositions.
A238130, A238279, A333755 count compositions by number of runs.
A274174 counts contiguous compositions, ranks A374249.
A335456 counts patterns matched by compositions.
A335548 counts non-contiguous compositions, ranks A374253.
Sequence in context: A090290 A153585 A169611 * A242460 A144494 A136166
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Jul 23 2024
STATUS
approved