cp's OEIS Frontend

A386576 Number of anti-runs of length n covering an initial interval of positive integers with strictly decreasing multiplicities.

%I A386576 #8 Aug 05 2025 21:33:32
%S A386576 1,1,0,1,0,1,10,4,14,84,1136,967,3342,12823,101762,1769580
%N A386576 Number of anti-runs of length n covering an initial interval of positive integers with strictly decreasing multiplicities.
%C A386576 An anti-run is a sequence with no adjacent equal terms.
%e A386576 The a(7) = 4 anti-runs are:
%e A386576   (1,2,1,2,1,2,1)
%e A386576   (1,2,1,2,1,3,1)
%e A386576   (1,2,1,3,1,2,1)
%e A386576   (1,3,1,2,1,2,1)
%t A386576 seps[ptn_,fir_]:=If[Total[ptn]==1,{{fir}},Join@@Table[Prepend[#,fir]&/@seps[MapAt[#-1&,ptn,fir],nex],{nex,Select[DeleteCases[Range[Length[ptn]],fir],ptn[[#]]>0&]}]];
%t A386576 seps[ptn_]:=If[Total[ptn]==0,{{}},Join@@(seps[ptn,#]&/@Range[Length[ptn]])];
%t A386576 Table[Sum[Length[seps[y]],{y,Select[IntegerPartitions[n],UnsameQ@@#&]}],{n,0,10}]
%Y A386576 For any multiplicities we have A005649.
%Y A386576 For weakly instead of strictly decreasing multiplicities we have A321688.
%Y A386576 A003242 and A335452 count anti-runs, ranks A333489.
%Y A386576 A005651 counts ordered set partitions with weakly decreasing sizes, strict A007837.
%Y A386576 A032020 counts strict anti-run compositions.
%Y A386576 A325534 counts separable multisets, ranks A335433.
%Y A386576 A325535 counts inseparable multisets, ranks A335448.
%Y A386576 A336103 counts normal separable multisets, inseparable A336102.
%Y A386576 A386583 counts separable partitions by length, inseparable A386584.
%Y A386576 A386585 counts partitions of separable type by length, sums A336106, ranks A335127.
%Y A386576 A386586 counts partitions of inseparable type by length, sums A025065, ranks A335126.
%Y A386576 A386633 counts separable set partitions, row sums of A386635.
%Y A386576 A386634 counts inseparable set partitions, row sums of A386636.
%Y A386576 Cf. A000670, A019472, A106351, A238130, A335125, A335516, A386580, A386581.
%K A386576 nonn,more
%O A386576 0,7
%A A386576 _Gus Wiseman_, Aug 03 2025