A316245 - OEIS

%I #50 Jan 19 2024 10:55:13

%S 1,1,3,6,14,25,52,89,167,279,486,786,1322,2069,3326,5128,8004,12055,

%T 18384,27203,40588,59186,86645,124583,179784,255111,362767,509319,

%U 715422,993681,1380793,1899630,2613064,3564177,4857631,6572314,8884973,11930363,16002853

%N Number of ways to split an integer partition of n into consecutive subsequences with weakly decreasing sums.

%H Andrew Howroyd, <a href="/A316245/b316245.txt">Table of n, a(n) for n = 0..50</a>

%H Gus Wiseman, <a href="/A316245/a316245.png">The a(5) = 25 split partitions.</a>

%H Gus Wiseman, <a href="/A316245/a316245_1.png">The a(6) = 52 split partitions.</a>

%e The a(4) = 14 split partitions:

%e   (4)

%e   (31)

%e   (22)

%e   (211)

%e   (3)(1)

%e   (2)(2)

%e   (1111)

%e   (21)(1)

%e   (2)(11)

%e   (111)(1)

%e   (11)(11)

%e   (2)(1)(1)

%e   (11)(1)(1)

%e   (1)(1)(1)(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],OrderedQ[Total/@#,GreaterEqual]&]],{y,IntegerPartitions[n]}],{n,10}]

%o (PARI) a(n)={my(recurse(r,m,s,t,f)=if(m==0, r==0, if(f, self()(r,min(m,t),t,0,0)) + self()(r,m-1,s,t,0) + if(t+m<=s, self()(r-m,min(m,r-m),s,t+m,1)))); recurse(n,n,n,0,0)} \\ _Andrew Howroyd_, Jan 18 2024

%Y Cf. A000041, A001970, A063834, A316223, A317546, A319001, A319003.

%Y Cf. A317508, A317715, A318434, A318683, A318684, A319794.

%K nonn

%O 0,3

%A _Gus Wiseman_, Sep 29 2018

%E a(21) onwards from _Andrew Howroyd_, Jan 18 2024