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%I #14 Sep 29 2018 12:56:55
%S 1,1,1,2,1,2,1,3,2,2,1,4,1,2,2,5,1,3,1,4,2,2,1,6,2,2,3,4,1,4,1,7,2,2,
%T 2,6,1,2,2,7,1,4,1,4,3,2,1,10,2,3,2,4,1,5,2,7,2,2,1,7,1,2,4,11,2,4,1,
%U 4,2,4,1,9,1,2,3,4,2,4,1,11,5,2,1,8,2,2
%N Number of ways to split the integer partition with Heinz number n into consecutive subsequences with weakly decreasing sums.
%C The Heinz number of an integer partition (y_1, ..., y_k) is prime(y_1) * ... * prime(y_k).
%e The a(60) = 7 split partitions:
%e (3)(2)(1)(1)
%e (32)(1)(1)
%e (3)(21)(1)
%e (3)(2)(11)
%e (321)(1)
%e (32)(11)
%e (3211)
%t comps[q_]:=Table[Table[Take[q,{Total[Take[c,i-1]]+1,Total[Take[c,i]]}],{i,Length[c]}],{c,Join@@Permutations/@IntegerPartitions[Length[q]]}];
%t Table[Length[Select[compositionPartitions[If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]],OrderedQ[Total/@#]&]],{n,100}]
%Y Cf. A001970, A056239, A063834, A255397, A296150, A316223, A317545, A317546, A319002, A319004.
%Y Cf. A316245, A317715, A318434, A318683, A318684, A319794.
%K nonn
%O 1,4
%A _Gus Wiseman_, Sep 29 2018
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