A374743 Number of integer compositions of n whose leaders of weakly decreasing runs are distinct.

1, 1, 2, 4, 8, 15, 29, 55, 105, 198, 371, 690, 1280, 2364, 4353, 7981, 14568, 26466, 47876, 86264, 154896, 277236, 494675, 879924, 1560275, 2757830, 4859010, 8534420, 14945107, 26096824, 45446624, 78939432, 136773519, 236401194, 407614349, 701147189, 1203194421

Offset: 0

Author

Gus Wiseman, Jul 25 2024

Keywords

• nonn

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.

Examples

			The composition (1,3,1,4,1,2,2,1) has maximal weakly decreasing subsequences ((1),(3,1),(4,1),(2,2,1)), with leaders (1,3,4,2), so is counted under a(15).
The a(0) = 1 through a(5) = 15 compositions:
  ()  (1)  (2)   (3)    (4)     (5)
           (11)  (12)   (13)    (14)
                 (21)   (22)    (23)
                 (111)  (31)    (32)
                        (112)   (41)
                        (121)   (113)
                        (211)   (122)
                        (1111)  (131)
                                (221)
                                (311)
                                (1112)
                                (1121)
                                (1211)
                                (2111)
                                (11111)
		

Links

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

Crossrefs

Ranked by A374701 = positions of distinct rows in A374740, opposite A374629.

Types of runs (instead of weakly decreasing):

- For leaders of identical runs we have A274174, ranks A374249.

- For leaders of anti-runs we have A374518, ranks A374638.

- For leaders of weakly increasing runs we have A374632, ranks A374768.

- For leaders of strictly increasing runs we have A374687, ranks A374698.

- For leaders of strictly decreasing runs we have A374761, ranks A374767.

Types of run-leaders (instead of distinct):

- For weakly increasing leaders we appear to have A188900.

- For identical leaders we have A374742.

- For strictly increasing leaders we have opposite A374634.

- For strictly decreasing leaders we have A374746.

- For weakly decreasing leaders we have A374747.

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.

A374748 counts compositions by sum of leaders of weakly decreasing runs.

Cf. A000009, A003242, A106356, A188920, A189076, A238343, A261982, A333213, A374635, A374741.

Programs

Mathematica

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

Extensions

a(24)-a(36) from Alois P. Heinz, Jul 26 2024