A384176 Number of subsets of {1..n} without all distinct lengths of maximal runs (increasing by 1).

0, 0, 0, 1, 3, 8, 20, 51, 121, 276, 612, 1335, 2881, 6144, 12950, 27029, 55977, 115222, 236058, 481683, 979443

Offset: 0

Author

Gus Wiseman, Jun 16 2025

Keywords

• nonn
• more

Examples

			The subset {1,3,4,8,9} has maximal runs ((1),(3,4),(8,9)), with lengths (1,2,2), so is counted under a(10).
The a(0) = 0 through a(6) = 20 subsets:
  .  .  .  {1,3}  {1,3}  {1,3}      {1,3}
                  {1,4}  {1,4}      {1,4}
                  {2,4}  {1,5}      {1,5}
                         {2,4}      {1,6}
                         {2,5}      {2,4}
                         {3,5}      {2,5}
                         {1,3,5}    {2,6}
                         {1,2,4,5}  {3,5}
                                    {3,6}
                                    {4,6}
                                    {1,3,5}
                                    {1,3,6}
                                    {1,4,6}
                                    {2,4,6}
                                    {1,2,4,5}
                                    {1,2,4,6}
                                    {1,2,5,6}
                                    {1,3,4,6}
                                    {1,3,5,6}
                                    {2,3,5,6}
		

Crossrefs

For equal instead of distinct lengths the complement is A243815.

These subsets are ranked by the non-members of A328592.

The complement is counted by A384175.

For strict partitions instead of subsets see A384178, A384884, A384886, A384880.

For permutations instead of subsets see A384891, A384892, A010027.

A034839 counts subsets by number of maximal runs, for strict partitions A116674.

A098859 counts Wilf partitions (distinct multiplicities), complement A336866.

A384893 counts subsets by number of maximal anti-runs, for partitions A268193, A384905.

Cf. A000009, A044813, A242882, A325325, A329739, A351202, A383013, A384177, A384889, A384890.

Programs

Mathematica

Table[Length[Select[Subsets[Range[n]],!UnsameQ@@Length/@Split[#,#2==#1+1&]&]],{n,0,10}]