A329749 Number of complete compositions of n whose run-lengths cover an initial interval of positive integers.

1, 1, 0, 2, 3, 5, 11, 23, 40, 80, 180, 344, 661, 1321, 2657, 5268, 10481, 20903, 41572, 82734, 164998, 328304, 654510, 1305421, 2598811, 5182174, 10332978, 20594318, 41066611, 81897091, 163309679, 325707492, 649648912, 1295827380, 2584941276, 5156774487

Offset: 0

Author

Gus Wiseman, Nov 21 2019

Keywords

• nonn

Comments

A composition of n is a finite sequence of positive integers with sum n. It is complete if it covers an initial interval of positive integers.

Examples

			The a(0) = 1 through a(6) = 11 compositions (empty column not shown):
  ()  (1)  (1,2)  (1,1,2)  (1,2,2)    (1,2,3)
           (2,1)  (1,2,1)  (2,1,2)    (1,3,2)
                  (2,1,1)  (2,2,1)    (2,1,3)
                           (1,1,2,1)  (2,3,1)
                           (1,2,1,1)  (3,1,2)
                                      (3,2,1)
                                      (1,2,1,2)
                                      (1,2,2,1)
                                      (2,1,1,2)
                                      (2,1,2,1)
                                      (1,1,2,1,1)
		

Crossrefs

Looking at multiplicities instead of run-lengths gives A329748.

The non-complete version is A329766.

Complete compositions are A107429.

Cf. A000740, A008965, A098504, A244164, A274174, A329738, A329739, A329740, A329741, A329744.

Programs

Mathematica

normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],normQ[#]&&normQ[Length/@Split[#]]&]],{n,0,10}]

Extensions

a(21)-a(35) from Alois P. Heinz, Jul 06 2020