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A268192 Triangle read by rows: T(n,k) is the number of partitions of weight k among the complements of the partitions of n. 2
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 (list; graph; refs; listen; history; text; internal format)
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.

Triangle starts:

1;

2;

2,1;

3,0,2;

2,2,0,2,1;

4,0,2,1,2,0,2;

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-Fran├žois Alcover, Dec 22 2016, after Alois P. Heinz *)

CROSSREFS

Cf. A000005, A000041, A033638, A070824, A188814.

Sequence in context: A124839 A294076 A117046 * A077653 A077889 A305805

Adjacent sequences:  A268189 A268190 A268191 * A268193 A268194 A268195

KEYWORD

nonn,tabf

AUTHOR

Emeric Deutsch, Feb 12 2016

STATUS

approved

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Last modified April 24 22:26 EDT 2019. Contains 322446 sequences. (Running on oeis4.)