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

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(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)
		

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.

Programs

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}]

Extensions

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