A385214 Number of subsets of {1..n} without all equal lengths of maximal runs of consecutive elements increasing by 1.

0, 0, 0, 0, 2, 8, 25, 66, 159, 361, 791, 1688, 3539, 7328, 15040, 30669, 62246, 125896, 253975, 511357, 1028052

Offset: 0

Author

Gus Wiseman, Jun 25 2025

Keywords

• nonn
• more

Examples

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

Crossrefs

These subsets are ranked by A164708, complement A164707

The complement is counted by A243815.

For distinct instead of equal lengths we have A384176, complement A384175.

For anti-runs instead of runs we have complement of A384889, for partitions A384888.

For permutations instead of subsets we have complement of A384892, distinct A384891.

For partitions instead of subsets we have complement of A384904, strict A384886.

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

A049988 counts partitions with equal run-lengths, distinct A325325.

A329738 counts compositions with equal run-lengths, distinct A329739.

A384177 counts subsets with all distinct lengths of maximal anti-runs, ranks A384879.

A384887 counts partitions with equal lengths of gapless runs, distinct A384884.

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

Cf. A010027, A044813, A164710, A383013, A384885.

Programs

Mathematica

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