A246721 Number of partitions of n into parts of the n-th list of distinct parts in the order given by A246688.

1, 1, 1, 2, 0, 2, 0, 2, 2, 0, 14, 3, 4, 0, 20, 3, 2, 1, 0, 26, 24, 4, 4, 2, 1, 35, 31, 4, 24, 2, 6, 1, 0, 378, 54, 42, 42, 5, 31, 0, 2, 0, 0, 631, 78, 61, 56, 5, 45, 34, 3, 3, 2, 2, 0, 1045, 992, 110, 85, 75, 73, 6, 55, 0, 7, 42, 8, 0, 2, 0, 1772, 1581, 156

Offset: 0

Author

Alois P. Heinz, Sep 02 2014

Keywords

• nonn
• look

Comments

The first lists of distinct parts in the order given by A246688 are: 0:[], 1:[1], 2:[2], 3:[1,2], 4:[3], 5:[1,3], 6:[4], 7:[1,4], 8:[2,3], 9:[5], 10:[1,2,3], 11:[1,5], 12:[2,4], 13:[6], 14:[1,2,4], 15:[1,6], 16:[2,5], 17:[3,4], 18:[7], 19:[1,2,5], 20:[1,3,4], ... .

Examples

			a(7) = 2 because there are 2 partitions of 7 into parts 1, 4: [1,1,1,1,1,1,1], [1,1,1,4].
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..5000

Crossrefs

Main diagonal of A246720.

Cf. A246688, A246691 (the same for compositions).

Programs

Maple

b:= proc(n, i) b(n, i):= `if`(n=0, [[]], `if`(i>n, [],
      [map(x->[i, x[]], b(n-i, i+1))[], b(n, i+1)[]]))
    end:
f:= proc() local i, l; i, l:=0, [];
      proc(n) while n>=nops(l)
        do l:=[l[], b(i, 1)[]]; i:=i+1 od; l[n+1]
      end
    end():
g:= proc(n, l) option remember; `if`(n=0, 1, `if`(l=[], 0,
      add(g(n-l[-1]*j, subsop(-1=NULL, l)), j=0..n/l[-1])))
    end:
a:= n-> g(n, f(n)):
seq(a(n), n=0..80);

Mathematica

b[n_, i_] := b[n, i] = If[n == 0, {{}}, If[i > n, {}, Join[Prepend[#, i]& /@ b[n - i, i + 1], b[n, i + 1]]]];
f = Module[{i = 0, l = {}}, Function[n, While[ n >= Length[l], l = Join[l, b[i, 1]]; i++]; l[[n + 1]]]];
g[n_, l_] := g[n, l] = If[n == 0, 1, If[l == {}, 0, Sum[g[n - l[[-1]] j, ReplacePart[l, -1 -> Nothing]], {j, 0, n/l[[-1]]}]]];
a[n_] := g[n, f[n]];
a /@ Range[0, 80] (* Jean-François Alcover, Dec 07 2020, after Alois P. Heinz *)

Formula

a(n) = A246720(n,n).