A375135 Number of integer compositions of n whose leaders of maximal strictly increasing runs are not weakly decreasing.

0, 0, 0, 0, 0, 1, 3, 9, 25, 63, 152, 355, 809, 1804, 3963, 8590, 18423, 39161, 82620, 173198, 361101, 749326, 1548609, 3189132, 6547190, 13404613

Offset

0,7

Comments

The leaders of maximal strictly increasing runs in a sequence are obtained by splitting it into maximal strictly increasing subsequences and taking the first term of each.

Links

Table of n, a(n) for n=0..25.

Gus Wiseman, Sequences counting and ranking compositions by their leaders (for six types of runs).

Example

The composition y = (1,2,1,3,2,3) has strictly increasing runs ((1,2),(1,3),(2,3)), with leaders (1,1,2), which are not weakly decreasing, so y is counted under a(12).
The a(0) = 0 through a(8) = 25 compositions:
  .  .  .  .  .  (122)  (132)   (133)    (143)
                        (1122)  (142)    (152)
                        (1221)  (1132)   (233)
                                (1222)   (1133)
                                (1321)   (1142)
                                (2122)   (1223)
                                (11122)  (1232)
                                (11221)  (1322)
                                (12211)  (1331)
                                         (1421)
                                         (2132)
                                         (3122)
                                         (11132)
                                         (11222)
                                         (11321)
                                         (12122)
                                         (12212)
                                         (12221)
                                         (13211)
                                         (21122)
                                         (21221)
                                         (111122)
                                         (111221)
                                         (112211)
                                         (122111)

Mathematica

Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n], !GreaterEqual@@First/@Split[#, Less]&]], {n, 0, 15}]

Crossrefs

For leaders of constant runs we have A056823.

For leaders of weakly increasing runs we have A374636, complement A189076?

The complement is counted by A374697.

For leaders of anti-runs we have A374699, complement A374682.

Other functional neighbors: A188920, A374764, A374765.

A003242 counts anti-run compositions, ranks A333489.

A011782 counts compositions.

A238130, A238279, A333755 count compositions by number of runs.

A335456 counts patterns matched by compositions.

A373949 counts compositions by run-compressed sum, opposite A373951.

A374700 counts compositions by sum of leaders of strictly increasing runs.

Cf. A106356, A238343, A261982, A333213, A374632, A374679, A374683, A374689.

Sequence in context: A065971 A145127 A096260 * A292326 A195417 A295142

Adjacent sequences: A375132 A375133 A375134 * A375136 A375137 A375138

Keyword

nonn,more

Author

Gus Wiseman, Aug 06 2024

Status

approved