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Number of ways to split a strict integer partition of n into consecutive subsequences with strictly decreasing sums.
18

%I #14 Sep 29 2018 12:57:30

%S 1,1,1,3,3,5,8,11,14,20,28,35,48,61,79,105,129,162,208,257,318,404,

%T 489,600,732,896,1075,1315,1576,1895,2272,2715,3217,3851,4537,5377,

%U 6353,7484,8765,10314,12044,14079,16420,19114,22184,25818,29840,34528,39903,46030

%N Number of ways to split a strict integer partition of n into consecutive subsequences with strictly decreasing sums.

%e The a(9) = 20 split partitions:

%e (9)

%e (81) (8)(1)

%e (72) (7)(2)

%e (63) (6)(3)

%e (54) (5)(4)

%e (432) (43)(2) (4)(3)(2)

%e (621) (62)(1) (6)(2)(1) (6)(21)

%e (531) (53)(1) (5)(3)(1) (5)(31)

%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[Sum[Length[Select[comps[y],OrderedQ[Total/@#,Greater]&]],{y,Select[IntegerPartitions[n],UnsameQ@@#&]}],{n,30}]

%Y Cf. A001970, A063834, A316245, A317508, A317546, A317715, A318434, A318683, A319794.

%K nonn

%O 0,4

%A _Gus Wiseman_, Sep 29 2018