A336108 Number of compositions of 2*n with n maximal runs.

1, 2, 4, 14, 36, 99, 274, 813, 2278, 6692, 19206, 56687, 164416, 486052, 1422654, 4214023, 12408476, 36825663, 108926976, 323856358, 961177042, 2862551860, 8518115200, 25407468667, 75763113682, 226297498429, 675951314988, 2021528322571, 6046881759308, 18104307275968, 54219605813884

Offset

0,2

Links

Andrew Howroyd, Table of n, a(n) for n = 0..200

Formula

a(n) = A333755(2*n,n).

a(n) = [x^(2*n)*y^n] (1 - y)/(1 - y - y*Sum_{d>=1} (1-y)^d*x^d/(1 - x^d)). - Andrew Howroyd, Feb 02 2021

Example

The a(0) = 1 through a(3) = 14 compositions:
  ()  (2)    (1,3)    (1,2,3)
      (1,1)  (3,1)    (1,3,2)
             (1,1,2)  (1,4,1)
             (2,1,1)  (2,1,3)
                      (2,3,1)
                      (3,1,2)
                      (3,2,1)
                      (1,1,3,1)
                      (1,2,2,1)
                      (1,3,1,1)
                      (2,1,1,2)
                      (1,1,1,2,1)
                      (1,1,2,1,1)
                      (1,2,1,1,1)

Mathematica

Table[Length[Select[Join@@Permutations/@IntegerPartitions[2*n], Length[Split[#]]==n&]], {n, 0, 10}]

Prog

(PARI) a(n)={polcoef(polcoef((1 - y)/(1 - y - y*sum(d=1, 2*n, (1-y)^d*x^d/(1 - x^d) + O(x^(2*n+1)))),  2*n, x), n, y)} \\ Andrew Howroyd, Feb 02 2021

Crossrefs

A333755 has this as main diagonal n = 2*k.

A337504 is the version for anti-runs.

A337505 is the version for anti-run patterns.

A337564 is the version for patterns.

A003242 counts anti-run compositions.

A011782 counts compositions.

A106356 counts compositions by number of adjacent equal parts.

A124767 counts maximal runs in standard compositions.

A238343 counts compositions by descents.

A272919 ranks runs.

A333213 counts compositions by weak ascents.

A333769 gives run-lengths of standard compositions.

Cf. A066099, A124762, A235998, A238130, A238279, A333381, A333382, A333489.

Sequence in context: A000622 A088028 A327459 * A263739 A333892 A135960

Adjacent sequences: A336105 A336106 A336107 * A336109 A336110 A336111

Keyword

nonn

Author

Gus Wiseman, Sep 04 2020

Extensions

Terms a(11) and beyond from Andrew Howroyd, Feb 02 2021

Status

approved