OFFSET
1,2
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).
LINKS
Alois P. Heinz, Rows n = 1..70, flattened
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.
EXAMPLE
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)
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 *)
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Emeric Deutsch, Feb 12 2016
STATUS
approved