editing

approved

a(0), a(21)-a(37) from Alois P. Heinz, Nov 21 2019

1, 1, 1, 3, 6, 11, 14, 34, 52, 114, 225, 464, 539, 1183, 1963, 3753, 6120, 11207, 19808, 38254, 77194, 147906, 224853, 374216, 611081, 1099933, 2129347, 3336099, 5816094, 9797957, 17577710, 29766586, 53276392, 93139668, 163600815, 324464546, 637029845, 1010826499

1,3

0,4

nonn,more,new

approved

editing

proposed

approved

editing

proposed

Looking at run-lengths instead of multiplicities gives A329742A329766.

allocated for Gus WisemanNumber of compositions of n whose multiplicities cover an initial interval of positive integers.

1, 1, 3, 6, 11, 14, 34, 52, 114, 225, 464, 539, 1183, 1963, 3753, 6120, 11207, 19808, 38254, 77194

1,3

A composition of n is a finite sequence of positive integers with sum n.

The a(1) = 1 through a(6) = 14 compositions:

(1) (2) (3) (4) (5) (6)

(1,2) (1,3) (1,4) (1,5)

(2,1) (3,1) (2,3) (2,4)

(1,1,2) (3,2) (4,2)

(1,2,1) (4,1) (5,1)

(2,1,1) (1,1,3) (1,1,4)

(1,2,2) (1,2,3)

(1,3,1) (1,3,2)

(2,1,2) (1,4,1)

(2,2,1) (2,1,3)

(3,1,1) (2,3,1)

(3,1,2)

(3,2,1)

(4,1,1)

normQ[m_]:=Or[m=={}, Union[m]==Range[Max[m]]];

Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], normQ[Length/@Split[Sort[#]]]&]], {n, 20}]

Looking at run-lengths instead of multiplicities gives A329742.

The complete case is A329748.

Complete compositions are A107429.

Cf. A000740, A008965, A098504, A242882, A244164, A329738, A329739, A329740.

allocated

nonn,more

Gus Wiseman, Nov 20 2019

approved

editing

allocated for Gus Wiseman

allocated

approved