A363224 Number of integer compositions of n in which the least part appears more than once.

0, 1, 1, 5, 8, 21, 44, 94, 197, 416, 857, 1766, 3621, 7392, 15032, 30493, 61708, 124646, 251359, 506203, 1018279, 2046454, 4109534, 8246985, 16540791, 33160051, 66451484, 133122753, 266612828, 533839069, 1068701695, 2139110054, 4281063708, 8566862025

Offset: 1

Author

Gus Wiseman, Jun 04 2023

Keywords

• nonn

Comments

Also the number of multisets of length n covering an initial interval of positive integers with more than one co-mode.

Examples

			The a(1) = 0 through a(6) = 21 compositions:
  .  (11)  (111)  (22)    (113)    (33)
                  (112)   (131)    (114)
                  (121)   (311)    (141)
                  (211)   (1112)   (222)
                  (1111)  (1121)   (411)
                          (1211)   (1113)
                          (2111)   (1122)
                          (11111)  (1131)
                                   (1212)
                                   (1221)
                                   (1311)
                                   (2112)
                                   (2121)
                                   (2211)
                                   (3111)
                                   (11112)
                                   (11121)
                                   (11211)
                                   (12111)
                                   (21111)
                                   (111111)
		

Links

• John Tyler Rascoe, Table of n, a(n) for n = 1..1000

Crossrefs

The complement is counted by A105039.

For partitions instead of compositions we have A117989.

Row sums of columns k > 1 of A238342.

If all parts appear more than once we have A240085, for partitions A007690.

If the least part appears exactly twice we have A241862.

For greatest instead of least we have A363262, see triangle A238341.

A000041 counts integer partitions, strict A000009.

A032020 counts strict compositions.

A067029 gives last exponent in prime factorization, first A071178.

A261982 counts compositions with some part appearing more than once.

A362607 counts partitions with multiple modes, co-modes A362609.

A362608 counts partitions with a unique mode, co-mode A362610.

Cf. A002865, A008284, A097979, A243737.

Programs

Mathematica

Table[Length[Select[Join@@Permutations /@ IntegerPartitions[n],Count[#,Min@@#]>1&]],{n,15}]

PARI

C_x(N)={my(x='x+O('x^N), h=sum(i=1,N,(x^(2*i)*(x-1)^3)/((x^i+x-1)*(x^(i+1)+x-1)^2))); concat([0],Vec(h))}
C_x(35) \\ John Tyler Rascoe, Jul 06 2024

Formula

G.f.: Sum_{i>0} (x^(2*i) * (x-1)^3)/((x^i + x - 1)*(x^(i+1) + x - 1)^2). - John Tyler Rascoe, Jul 06 2024