A353391 Number of compositions of n that are empty, a singleton, or whose run-lengths are a subsequence that is already counted.

1, 1, 1, 1, 2, 1, 3, 1, 1, 4, 5, 7, 9, 11, 15, 22, 38, 45, 87, 93

Offset

0,5

Links

Table of n, a(n) for n=0..19.

Example

The a(9) = 4 through a(14) = 15 compositions (A..E = 10..14):
  (9)       (A)       (B)       (C)       (D)       (E)
  (333)     (2233)    (141122)  (2244)    (161122)  (2255)
  (121122)  (3322)    (221123)  (4422)    (221125)  (5522)
  (221121)  (131122)  (221132)  (151122)  (221134)  (171122)
            (221131)  (221141)  (221124)  (221143)  (221126)
                      (231122)  (221142)  (221152)  (221135)
                      (321122)  (221151)  (221161)  (221153)
                                (241122)  (251122)  (221162)
                                (421122)  (341122)  (221171)
                                          (431122)  (261122)
                                          (521122)  (351122)
                                                    (531122)
                                                    (621122)
                                                    (122121122)
                                                    (221121221)

Mathematica

yosQ[y_]:=Length[y]<=1||MemberQ[Subsets[y], Length/@Split[y]]&&yosQ[Length/@Split[y]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], yosQ]], {n, 0, 15}]

Crossrefs

The non-recursive version is A353390, ranked by A353402.

The non-recursive consecutive version is A353392, ranked by A353432.

The non-recursive reverse version is A353403.

The unordered version is A353426, ranked by A353393 (nonprime A353389).

The consecutive version is A353430.

These compositions are ranked by A353431.

A003242 counts anti-run compositions, ranked by A333489.

A011782 counts compositions.

A329738 counts uniform compositions, partitions A047966.

A114901 counts compositions with no runs of length 1.

A169942 counts Golomb rulers, ranked by A333222.

A325676 counts knapsack compositions, ranked by A333223.

A325705 counts partitions containing all of their distinct multiplicities.

A329739 counts compositions with all distinct run-length.

Cf. A005811, A032020, A103295, A114640, A165413, A181591, A242882, A324572, A325702, A333755, A351013, A353401.

Sequence in context: A294099 A209115 A353430 * A141412 A178623 A210765

Adjacent sequences: A353388 A353389 A353390 * A353392 A353393 A353394

Keyword

nonn,more

Author

Gus Wiseman, May 15 2022

Status

approved