A329748 Number of complete compositions of n whose multiplicities cover an initial interval of positive integers.

1, 1, 0, 2, 3, 3, 6, 12, 12, 42, 114, 210, 60, 360, 720, 1320, 1590, 3690, 6450, 16110, 33120, 59940, 61320, 112980, 171780, 387240, 803880, 769440, 1773240, 2823240, 5790960, 9916200, 19502280, 28244160, 56881440, 130548600, 279578880, 320554080, 541323720

Offset

0,4

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.

Links

Table of n, a(n) for n=0..38.

Example

The a(1) = 1 through a(8) = 12 compositions (empty column not shown):
  (1)  (12)  (112)  (122)  (123)  (1123)  (1223)
       (21)  (121)  (212)  (132)  (1132)  (1232)
             (211)  (221)  (213)  (1213)  (1322)
                           (231)  (1231)  (2123)
                           (312)  (1312)  (2132)
                           (321)  (1321)  (2213)
                                  (2113)  (2231)
                                  (2131)  (2312)
                                  (2311)  (2321)
                                  (3112)  (3122)
                                  (3121)  (3212)
                                  (3211)  (3221)

Mathematica

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

Crossrefs

Looking at run-lengths instead of multiplicities gives A329749.

The non-complete version is A329741.

Complete compositions are A107429.

Cf. A008965, A098504, A244164, A274174, A329740, A329744, A329766.

Sequence in context: A054630 A049875 A180887 * A173094 A087989 A028257

Adjacent sequences: A329745 A329746 A329747 * A329749 A329750 A329751

Keyword

nonn

Author

Gus Wiseman, Nov 21 2019

Extensions

a(21)-a(38) from Alois P. Heinz, Jul 06 2020

Status

approved