A268192 Triangle read by rows: T(n,k) is the number of partitions of weight k among the complements of the partitions of n.

1, 2, 2, 1, 3, 0, 2, 2, 2, 0, 2, 1, 4, 0, 2, 1, 2, 0, 2, 2, 2, 2, 2, 0, 4, 0, 0, 2, 1, 4, 1, 2, 0, 6, 0, 2, 2, 1, 0, 2, 0, 2, 3, 2, 0, 6, 0, 2, 4, 4, 0, 2, 0, 2, 2, 0, 0, 2, 1, 4, 0, 6, 0, 2, 4, 5, 0, 6, 0, 4, 2, 0, 0, 4, 1, 0, 0, 2, 0, 2, 2, 4, 0, 2, 6, 5, 0, 6, 0, 8

Offset: 1

Author

Emeric Deutsch, Feb 12 2016

Keywords

• nonn
• tabf

Comments

The complement of a partition p[1] >= p[2] >=...>= p[k] is p[1]-p[2], p[1]-p[3], ..., p[1]-p[k]. Its Ferrers board emerges naturally from the Ferrers board of the given partition. The weight of a partition of n is n.

Sum of entries in row n is A000041(n) (the partition numbers).

Apparently, number of entries in row n is A033638(n-1) = 1 + floor((n-1)^2/4).

T(n,0) = A000005(n) = number of divisors of n.

T(n,1) = A070824(n+1).

Sum(k*T(n,k),k>0) = A188814(n).

Examples

			Row 4 is 3,0,2; indeed, the complements of [4], [3,1], [2,2], [2,1,1], [1,1,1,1] are: empty, [2], empty, [1,1], empty; their weights are 0, 2, 0, 2, 0, respectively.
From _Gus Wiseman_, Sep 24 2019: (Start)
Triangle begins:
  1
  2
  2 1
  3 0 2
  2 2 0 2 1
  4 0 2 1 2 0 2
  2 2 2 2 0 4 0 0 2 1
  4 1 2 0 6 0 2 2 1 0 2 0 2
  3 2 0 6 0 2 4 4 0 2 0 2 2 0 0 2 1
  4 0 6 0 2 4 5 0 6 0 4 2 0 0 4 1 0 0 2 0 2
  2 4 0 2 6 5 0 6 0 8 4 0 0 6 2 0 2 2 0 2 0 2 0 0 2 1
Row  n = 8 counts the following partitions:
  8          332   53      62       71        521     4211   611      5111
  44               22211   422      2111111   32111          311111   41111
  2222                     431
  11111111                 3221
                           3311
                           221111
(End)
		

Links

• Alois P. Heinz, Rows n = 1..70, flattened

Crossrefs

Cf. A000005, A000041, A002620, A033638, A070824, A188814, A326844, A326849.

Programs

Maple

q := 10: with(combinat): a := proc (i, j) options operator, arrow: partition(i)[j] end proc: P[q] := 0: for j to numbpart(q) do P[q] := sort(P[q]+t^(nops(a(q, j))*max(a(q, j))-q)) end do: P[q] := P[q];
# second Maple program:
b:= proc(n, i, l) option remember; expand(`if`(n=0 or i=1,
      x^(`if`(l=0, 0, n*(l-i))), b(n, i-1, l)+`if`(i>n, 0,
      x^(`if`(l=0, 0, l-i))*b(n-i, i, `if`(l=0, i, l)))))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2, 0)):
seq(T(n), n=1..15);  # Alois P. Heinz, Feb 12 2016

Mathematica

b[n_, i_, l_] := b[n, i, l] = Expand[If[n == 0 || i == 1, x^(If[l == 0, 0, n*(l - i)]), b[n, i - 1, l] + If[i > n, 0, x^(If[l == 0, 0, l - i])*b[n - i, i, If[l == 0, i, l]]]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, n, 0]]; Table[T[n], {n, 1, 15}] // Flatten (* Jean-François Alcover, Dec 22 2016, after Alois P. Heinz *)
Table[Length[Select[IntegerPartitions[n],Max[#]*Length[#]-n==k&]],{n,1,11},{k,0,Floor[(n-1)/2]*Ceiling[(n-1)/2]}] (* Gus Wiseman, Sep 24 2019 *)

Formula

The weight of the complement of a partition p is (number of parts of p)*(largest part of p) - weight of p.

For a given q, the Maple program yields the generating polynomial of row q.