A237513 T(n,k) = number of maximal horizontal rectangles that contain the Durfee square for partitions of n that consist of k nodes, 1 <= k <= n; triangular array read by rows.

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1, 5, 1, 2, 1, 1, 1, 7, 1, 3, 1, 1, 1, 1, 10, 1, 5, 1, 2, 1, 1, 1, 13, 1, 7, 1, 3, 2, 1, 1, 1, 17, 1, 10, 1, 5, 3, 2, 1, 1, 1, 21, 1, 13, 1, 7, 6, 3, 1, 1, 1, 1, 26, 1, 17, 1, 10, 10, 5, 1, 3, 1, 1, 1, 31, 1

Offset

1,10

Comments

Column n consists solely of 1's if and only if n = 1 or n is a prime. Column 4: A033638.

Links

Clark Kimberling, Table of n, a(n) for n = 1..500

Example

Triangle begins:
  1
  1 ... 1
  1 ... 1 ... 1
  1 ... 1 ... 1 ... 3
  1 ... 1 ... 1 ... 3 ... 1
  1 ... 1 ... 1 ... 5 ... 1 ... 2
  1 ... 1 ... 1 ... 7 ... 1 ... 3 ... 1
  1 ... 1 ... 1 .. 10 ... 1 ... 5 ... 1 ... 2
  1 ... 1 ... 1 .. 13 ... 1 ... 7 ... 1 ... 3 ... 2
  1 ... 1 ... 1 .. 17 ... 1 .. 10 ... 1 ... 5 ... 3 ... 2
  1 ... 1 ... 1 .. 21 ... 1 .. 13 ... 1 ... 7 ... 6 ... 3 ... 1
  1 ... 1 ... 1 .. 26 ... 1 .. 17 ... 1 .. 10 .. 10 ... 5 ... 1 ... 3
The dimensions of maximal horizontal rectangles for the partitions of 6 are 1 X 6, 1 X 5, 2 X 2, 1 X 4, 2 X 3, 2 X 2, 1 X 3, 2 X 2, 2 X 2, 1 X 2, 1 X 1; the numbers of nodes are the products 6, 5, 4, 4, 6, 4, 3, 4, 4, 2, 1; counting the occurrences of 1,2,3,4,5,6 gives 1,1,1,5,1,2, which is row 6 of the array.

Mathematica

durfeeSquare[part_] := Max[Map[Min, Transpose[{#, Range[Length[#]]}] &[part]]]; hRectangle[part_] := {#, part[[#]]} &[durfeeSquare[part]]; Map[Last[Transpose[Tally[Sort[Map[Times @@ hRectangle[#] &, IntegerPartitions[#]]]]]] &, Range[20]] (* Peter J. C. Moses, Feb 10 2014 *)

Crossrefs

Cf. A033638, A000041, A033638.

Sequence in context: A094392 A111946 A175788 * A137844 A263845 A079229

Adjacent sequences: A237510 A237511 A237512 * A237514 A237515 A237516

Keyword

nonn,tabl,easy

Author

Clark Kimberling, Feb 08 2014

Status

approved