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approved

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proposed

A329739 counts compositions with all distinct run-lengths, for runs A351013.

Cf. A005811, A032020, A103295, A114640, A165413, A242882, A325705, A333755, A351013, A353400, A353401.

The nonNon-recursive non-consecutive version is : counted by A353390, ranked by A353402, reverse A353403, partitions A325702.

The nonNon-consecutive version is : A353391, ranked by A353431, partitions A353426.

The nonNon-recursive version is : A353392, ranked by A353432.

A353400 counts compositions with all run-lengths > 2.

Cf. A005811, A032020, A103295, A114640, A165413, A242882, A325705, A333755, A353400, A353401.

allocated for Gus WisemanNumber of integer compositions of n that are empty, a singleton, or whose own run-lengths are a consecutive subsequence that is already counted.

1, 1, 1, 1, 2, 1, 3, 1, 1, 4, 5, 7, 9, 11, 15, 16, 22, 25, 37, 37, 45

0,5

The a(n) compositions for selected n (A..E = 10..14):

n=4: n=6: n=9: n=10: n=12: n=14:

-----------------------------------------------------------

(4) (6) (9) (A) (C) (E)

(22) (1122) (333) (2233) (2244) (2255)

(2211) (121122) (3322) (4422) (5522)

(221121) (131122) (151122) (171122)

(221131) (221124) (221126)

(221142) (221135)

(221151) (221153)

(241122) (221162)

(421122) (221171)

(261122)

(351122)

(531122)

(621122)

(122121122)

(221121221)

yoyQ[y_]:=Length[y]<=1||MemberQ[Join@@Table[Take[y, {i, j}], {i, Length[y]}, {j, i, Length[y]}], Length/@Split[y]]&&yoyQ[Length/@Split[y]];

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], yoyQ]], {n, 0, 15}]

The non-recursive non-consecutive version is counted by A353390, ranked by A353402, reverse A353403, partitions A325702.

The non-consecutive version is A353391, ranked by A353431, partitions A353426.

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

A003242 counts anti-run compositions, ranked by A333489.

A011782 counts compositions.

A114901 counts compositions with no runs of length 1.

A169942 counts Golomb rulers, ranked by A333222.

A325676 counts knapsack compositions, ranked by A333223.

A329738 counts uniform compositions, partitions A047966.

A329739 counts compositions with all distinct run-lengths, for runs A351013.

A353400 counts compositions with all run-lengths > 2.

Cf. A005811, A032020, A103295, A114640, A165413, A242882, A325705, A333755, A353401.

allocated

nonn,more

Gus Wiseman, May 16 2022

approved

editing

allocated for Gus Wiseman

allocated

approved