A246690 -id:A246690 - cp's OEIS frontend

A246691 Number of compositions of n into parts of the n-th list of distinct parts in the order given by A246688.

1, 1, 1, 3, 0, 4, 0, 5, 4, 0, 274, 11, 13, 0, 1601, 21, 11, 10, 0, 15571, 7921, 53, 41, 12, 1, 246441, 64208, 119, 16169, 47, 89, 35, 0, 1439975216, 17319590, 1835123, 956698, 531, 274291, 0, 82, 0, 0, 428262742476, 1923714115, 72992449, 20086406, 1915, 4051405

Offset: 0

Alois P. Heinz, Sep 01 2014

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], ... .

			a(7) = 5 because there are 5 compositions of 7 into parts 1, 4: [1,1,1,1,1,1,1], [1,1,1,4], [1,1,4,1], [1,4,1,1], [4,1,1,1].

Alois P. Heinz, Table of n, a(n) for n = 0..1260

Cf. A246688, A246690, A246721 (the same for partitions).

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,
      add(`if`(i>n, 0, g(n-i, l)), i=l))
    end:
a:= n-> g(n, f(n)):
seq(a(n), n=0..80);

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, Sum[If[i>n, 0, g[n - i, l]], {i, l}]];
a[n_] := g[n, f[n]];
Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Jul 12 2021, after Alois P. Heinz *)

a(n) = A246690(n,n).

A246720 Number A(n,k) of partitions of n into parts of the k-th list of distinct parts in the order given by A246688; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 2, 0, 1, 0, 1, 1, 0, 2, 1, 1, 0, 1, 0, 1, 1, 3, 0, 1, 0, 1, 1, 0, 2, 0, 3, 1, 1, 0, 1, 0, 1, 0, 2, 0, 4, 0, 1, 0, 1, 0, 1, 1, 1, 2, 1, 4, 1, 1, 0, 1, 1, 0, 1, 2, 0, 3, 0, 5, 0, 1, 0, 1, 1, 2, 0, 1, 2, 0, 3, 0, 5, 1, 1, 0

Offset: 0

Alois P. Heinz, Sep 02 2014

			Square array A(n,k) begins:
  1, 1, 1, 1, 1, 1, 1, 1, 1, 1,  1, 1, 1, 1,  1, ...
  0, 1, 0, 1, 0, 1, 0, 1, 0, 0,  1, 1, 0, 0,  1, ...
  0, 1, 1, 2, 0, 1, 0, 1, 1, 0,  2, 1, 1, 0,  2, ...
  0, 1, 0, 2, 1, 2, 0, 1, 1, 0,  3, 1, 0, 0,  2, ...
  0, 1, 1, 3, 0, 2, 1, 2, 1, 0,  4, 1, 2, 0,  4, ...
  0, 1, 0, 3, 0, 2, 0, 2, 1, 1,  5, 2, 0, 0,  4, ...
  0, 1, 1, 4, 1, 3, 0, 2, 2, 0,  7, 2, 2, 1,  6, ...
  0, 1, 0, 4, 0, 3, 0, 2, 1, 0,  8, 2, 0, 0,  6, ...
  0, 1, 1, 5, 0, 3, 1, 3, 2, 0, 10, 2, 3, 0,  9, ...
  0, 1, 0, 5, 1, 4, 0, 3, 2, 0, 12, 2, 0, 0,  9, ...
  0, 1, 1, 6, 0, 4, 0, 3, 2, 1, 14, 3, 3, 0, 12, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns k=0-11, 13-20, 23 give: A000007, A000012, A059841, A008619, A079978, A008620, A121262, A008621, A103221, A079998, A001399, A002266(n+5), A079979, A008642, A097992, A008616, A008679, A082784, A000115, A025767, A008676.
Main diagonal gives A246721.
Cf. A246688, A246690 (the same for compositions).

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, k)-> g(n, f(k)):
seq(seq(A(n, d-n), n=0..d), d=0..16);

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_, k_] := g[n, f[k]];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 16}] // Flatten (* Jean-François Alcover, Dec 07 2020, after Alois P. Heinz *)

cp's OEIS Frontend

A246691 Number of compositions of n into parts of the n-th list of distinct parts in the order given by A246688.

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