A374763 Number of integer compositions of n whose leaders of strictly decreasing runs are themselves strictly decreasing.

1, 1, 1, 2, 3, 4, 6, 10, 15, 22, 32, 47, 71, 106, 156, 227, 328, 473, 683, 986, 1421, 2040, 2916, 4149

Offset

0,4

Comments

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

Links

Table of n, a(n) for n=0..23.

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

Example

The composition (3,1,2,1,1) has strictly decreasing runs ((3,1),(2,1),(1)), with leaders (3,2,1), so is counted under a(8).
The a(0) = 1 through a(8) = 15 compositions:
  ()  (1)  (2)  (3)   (4)    (5)    (6)    (7)     (8)
                (21)  (31)   (32)   (42)   (43)    (53)
                      (211)  (41)   (51)   (52)    (62)
                             (311)  (312)  (61)    (71)
                                    (321)  (322)   (413)
                                    (411)  (412)   (422)
                                           (421)   (431)
                                           (511)   (512)
                                           (3121)  (521)
                                           (3211)  (611)
                                                   (3212)
                                                   (3221)
                                                   (4121)
                                                   (4211)
                                                   (31211)

Mathematica

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

Crossrefs

The opposite version is A374688.

The weak version is A374747.

For partitions instead of compositions we have A375133.

Other types of runs (instead of strictly decreasing):

- For leaders of identical runs we have A000041.

- For leaders of weakly increasing runs we appear to have A188920.

- For leaders of anti-runs we have A374680.

- For leaders of strictly increasing runs we have A374689.

- For leaders of weakly decreasing runs we have A374746.

Other types of run-leaders (instead of strictly decreasing):

- For identical leaders we have A374760, ranks A374759.

- For distinct leaders we have A374761, ranks A374767.

- For strictly increasing leaders we have A374762.

- For weakly increasing leaders we have A374764.

- For weakly decreasing leaders we have A374765.

A003242 counts anti-run compositions, ranks A333489.

A011782 counts compositions.

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

A274174 counts contiguous compositions, ranks A374249.

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

Cf. A000009, A106356, A188900, A238343, A261982, A333213, A374518, A374632, A374635, A374687, A374742, A374743.

Sequence in context: A086990 A090412 A073028 * A147788 A104977 A206742

Adjacent sequences: A374760 A374761 A374762 * A374764 A374765 A374766

Keyword

nonn,more

Author

Gus Wiseman, Jul 30 2024

Status

approved