A384177 Number of subsets of {1..n} with all distinct lengths of maximal anti-runs (increasing by more than 1).

1, 2, 3, 5, 10, 19, 35, 62, 109, 197, 364, 677, 1251, 2288, 4143, 7443, 13318, 23837, 42809, 77216, 139751, 253293, 458800, 829237, 1494169, 2683316, 4804083, 8580293, 15301324, 27270061, 48607667, 86696300, 154758265, 276453311, 494050894, 882923051

Offset

0,2

Links

Christian Sievers, Table of n, a(n) for n = 0..1000

Example

The subset {1,2,4,5,7,10} has maximal anti-runs ((1),(2,4),(5,7,10)), with lengths (1,2,3), so is counted under a(10).
The a(0) = 1 through a(5) = 19 subsets:
  {}  {}   {}   {}     {}       {}
      {1}  {1}  {1}    {1}      {1}
           {2}  {2}    {2}      {2}
                {3}    {3}      {3}
                {1,3}  {4}      {4}
                       {1,3}    {5}
                       {1,4}    {1,3}
                       {2,4}    {1,4}
                       {1,2,4}  {1,5}
                       {1,3,4}  {2,4}
                                {2,5}
                                {3,5}
                                {1,2,4}
                                {1,2,5}
                                {1,3,4}
                                {1,3,5}
                                {1,4,5}
                                {2,3,5}
                                {2,4,5}

Mathematica

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

Prog

(PARI) lista(n)={my(o=(1-x^(n+1))/(1-x)*O(y*y^n), p=prod(i=1, (n+1)\2, 1+o+x*y^(2*i-1)/(1-y)^(i-1))); p=subst(serlaplace(p), x, 1); Vec((p-y)/(1-y)^2)} \\ Christian Sievers, Jun 18 2025

Crossrefs

For runs instead of anti-runs we have A384175, complement A384176.

These subsets are ranked by A384879.

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

For equal instead of distinct lengths we have A384889, for runs A243815.

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, A010027, A044813, A047993, A106529, A123513, A242882, A325325, A328592, A329739, A351202, A384890.

Sequence in context: A283314 A078715 A166874 * A390560 A046630 A177874

Adjacent sequences: A384174 A384175 A384176 * A384178 A384179 A384180

Keyword

nonn

Author

Gus Wiseman, Jun 16 2025

Extensions

a(21) and beyond from Christian Sievers, Jun 18 2025

Status

approved