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Number of ways to split a strict integer partition of n into consecutive subsequences with equal sums.
16

%I #16 Sep 29 2018 12:57:23

%S 1,1,1,2,2,3,5,5,7,8,12,12,18,18,26,27,37,38,53,54,73,76,100,104,136,

%T 142,183,192,244,256,327,340,424,448,558,585,722,760,937,983,1195,

%U 1260,1544,1610,1943,2053,2480,2590,3107,3264,3927,4106,4874,5120,6134,6378

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

%e The a(12) = 18 constant-sum split partitions:

%e (12)

%e (7,5)

%e (8,4)

%e (9,3)

%e (10,2)

%e (11,1)

%e (5,4,3)

%e (6,4,2)

%e (6,5,1)

%e (7,3,2)

%e (7,4,1)

%e (8,3,1)

%e (9,2,1)

%e (6)(4,2)

%e (6)(5,1)

%e (5,4,2,1)

%e (6,3,2,1)

%e (6)(3,2,1)

%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],SameQ@@Total/@#&]],{y,Select[IntegerPartitions[n],UnsameQ@@#&]}],{n,30}]

%Y Cf. A001970, A063834, A316245, A317508, A317546, A317715, A318434, A318683, A318684, A319794.

%K nonn

%O 0,4

%A _Gus Wiseman_, Sep 29 2018