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, 1985518, 4014702, 8100539, 16316663, 32821062, 65946508, 132384357, 265556835, 532369056, 1066724980, 2136578199, 4278062897, 8563802132, 17139590076

Offset

0,5

Links

Table of n, a(n) for n=0..34.

Example

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}

Mathematica

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

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.

Sequence in context: A140662 A174198 A396787 * A077997 A294407 A295346

Adjacent sequences: A384173 A384174 A384175 * A384177 A384178 A384179

Keyword

nonn

Author

Gus Wiseman, Jun 16 2025

Extensions

a(21) onward from A384175 by Martin Fuller, Feb 23 2026

Status

approved