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
Keywords
Examples
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)
Links
- Andrew Howroyd, Table of n, a(n) for n = 0..200
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.
Programs
-
Mathematica
Table[Length[Select[Join@@Permutations/@IntegerPartitions[2*n],Length[Split[#]]==n&]],{n,0,10}] -
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
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
Extensions
Terms a(11) and beyond from Andrew Howroyd, Feb 02 2021