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Number of compositions of 2*n with n maximal anti-runs.
4

%I #16 Feb 02 2021 18:26:01

%S 1,1,3,8,13,33,112,286,769,2288,6695,18745,54654,160888,467402,

%T 1362378,4016517,11807966,34708018,102451390,302870005,895207191,

%U 2650590597,7859253320,23316653154,69231883374,205773157904,612021943421,1821435719846,5424528040529,16165017705176

%N Number of compositions of 2*n with n maximal anti-runs.

%C An anti-run is a sequence with no adjacent equal parts.

%H Andrew Howroyd, <a href="/A337504/b337504.txt">Table of n, a(n) for n = 0..200</a>

%F a(n) = [x^(2*n)*y^n] 1 - y + y*(y-1)/(y - 1 - Sum_{d>=1} (y-1)^d*x^d/(1 - x^d)). - _Andrew Howroyd_, Feb 02 2021

%e The a(0) = 1 through a(4) = 13 compositions:

%e () (2) (2,2) (2,2,2) (2,2,2,2)

%e (1,1,2) (1,1,1,3) (1,1,1,1,4)

%e (2,1,1) (1,1,2,2) (1,1,2,2,2)

%e (2,2,1,1) (2,2,2,1,1)

%e (3,1,1,1) (4,1,1,1,1)

%e (1,1,1,2,1) (1,1,1,1,3,1)

%e (1,1,2,1,1) (1,1,1,2,2,1)

%e (1,2,1,1,1) (1,1,1,3,1,1)

%e (1,1,2,2,1,1)

%e (1,1,3,1,1,1)

%e (1,2,2,1,1,1)

%e (1,3,1,1,1,1)

%e (2,1,1,1,1,2)

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

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

%Y A106356 has this as main diagonal n = 2*k.

%Y A336108 is the version for runs.

%Y A337505 is the version for patterns.

%Y A337564 is the version for runs in patterns.

%Y A003242 counts anti-run compositions.

%Y A011782 counts compositions.

%Y A124767 counts runs in standard compositions.

%Y A238343 counts compositions by descents.

%Y A333213 counts compositions by weak ascents.

%Y A333381 counts anti-runs in standard compositions.

%Y A333382 counts adjacent unequal pairs in standard compositions.

%Y A333489 ranks anti-runs.

%Y A333755 counts compositions by number of runs.

%Y A333769 gives run-lengths in standard compositions.

%Y A337565 gives anti-run lengths in standard compositions.

%Y Cf. A106351, A124762, A233564, A235998, A238130, A238279, A333214, A333216.

%K nonn

%O 0,3

%A _Gus Wiseman_, Sep 04 2020

%E Terms a(11) and beyond from _Andrew Howroyd_, Feb 02 2021