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A337504 Number of compositions of 2*n with n maximal anti-runs.

%I A337504 #16 Feb 02 2021 18:26:01
%S A337504 1,1,3,8,13,33,112,286,769,2288,6695,18745,54654,160888,467402,
%T A337504 1362378,4016517,11807966,34708018,102451390,302870005,895207191,
%U A337504 2650590597,7859253320,23316653154,69231883374,205773157904,612021943421,1821435719846,5424528040529,16165017705176
%N A337504 Number of compositions of 2*n with n maximal anti-runs.
%C A337504 An anti-run is a sequence with no adjacent equal parts.
%H A337504 Andrew Howroyd, <a href="/A337504/b337504.txt">Table of n, a(n) for n = 0..200</a>
%F A337504 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 A337504 The a(0) = 1 through a(4) = 13 compositions:
%e A337504   ()  (2)  (2,2)    (2,2,2)      (2,2,2,2)
%e A337504            (1,1,2)  (1,1,1,3)    (1,1,1,1,4)
%e A337504            (2,1,1)  (1,1,2,2)    (1,1,2,2,2)
%e A337504                     (2,2,1,1)    (2,2,2,1,1)
%e A337504                     (3,1,1,1)    (4,1,1,1,1)
%e A337504                     (1,1,1,2,1)  (1,1,1,1,3,1)
%e A337504                     (1,1,2,1,1)  (1,1,1,2,2,1)
%e A337504                     (1,2,1,1,1)  (1,1,1,3,1,1)
%e A337504                                  (1,1,2,2,1,1)
%e A337504                                  (1,1,3,1,1,1)
%e A337504                                  (1,2,2,1,1,1)
%e A337504                                  (1,3,1,1,1,1)
%e A337504                                  (2,1,1,1,1,2)
%t A337504 Table[Length[Select[Join@@Permutations/@IntegerPartitions[2*n],Length[Split[#,UnsameQ]]==n&]],{n,0,10}]
%o A337504 (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 A337504 A106356 has this as main diagonal n = 2*k.
%Y A337504 A336108 is the version for runs.
%Y A337504 A337505 is the version for patterns.
%Y A337504 A337564 is the version for runs in patterns.
%Y A337504 A003242 counts anti-run compositions.
%Y A337504 A011782 counts compositions.
%Y A337504 A124767 counts runs in standard compositions.
%Y A337504 A238343 counts compositions by descents.
%Y A337504 A333213 counts compositions by weak ascents.
%Y A337504 A333381 counts anti-runs in standard compositions.
%Y A337504 A333382 counts adjacent unequal pairs in standard compositions.
%Y A337504 A333489 ranks anti-runs.
%Y A337504 A333755 counts compositions by number of runs.
%Y A337504 A333769 gives run-lengths in standard compositions.
%Y A337504 A337565 gives anti-run lengths in standard compositions.
%Y A337504 Cf. A106351, A124762, A233564, A235998, A238130, A238279, A333214, A333216.
%K A337504 nonn
%O A337504 0,3
%A A337504 _Gus Wiseman_, Sep 04 2020
%E A337504 Terms a(11) and beyond from _Andrew Howroyd_, Feb 02 2021