A329741 Number of compositions of n whose multiplicities cover an initial interval of positive integers.

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

Offset

0,4

Comments

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

Links

Table of n, a(n) for n=0..37.

Example

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)

Mathematica

normQ[m_]:=Or[m=={}, Union[m]==Range[Max[m]]];
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n], normQ[Length/@Split[Sort[#]]]&]], {n, 20}]

Crossrefs

Looking at run-lengths instead of multiplicities gives A329766.

The complete case is A329748.

Complete compositions are A107429.

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

Sequence in context: A015823 A049620 A350438 * A310096 A310097 A310098

Adjacent sequences: A329738 A329739 A329740 * A329742 A329743 A329744

Keyword

nonn

Author

Gus Wiseman, Nov 20 2019

Extensions

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

Status

approved