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A316245
Number of ways to split an integer partition of n into consecutive subsequences with weakly decreasing sums.
27
1, 1, 3, 6, 14, 25, 52, 89, 167, 279, 486, 786, 1322, 2069, 3326, 5128, 8004, 12055, 18384, 27203, 40588, 59186, 86645, 124583, 179784, 255111, 362767, 509319, 715422, 993681, 1380793, 1899630, 2613064, 3564177, 4857631, 6572314, 8884973, 11930363, 16002853
OFFSET
0,3
EXAMPLE
The a(4) = 14 split partitions:
(4)
(31)
(22)
(211)
(3)(1)
(2)(2)
(1111)
(21)(1)
(2)(11)
(111)(1)
(11)(11)
(2)(1)(1)
(11)(1)(1)
(1)(1)(1)(1)
MATHEMATICA
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]]}];
Table[Sum[Length[Select[comps[y], OrderedQ[Total/@#, GreaterEqual]&]], {y, IntegerPartitions[n]}], {n, 10}]
PROG
(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
KEYWORD
nonn
AUTHOR
Gus Wiseman, Sep 29 2018
EXTENSIONS
a(21) onwards from Andrew Howroyd, Jan 18 2024
STATUS
approved