A115724 -id:A115724 - cp's OEIS frontend

A115723 Table of partitions of n with maximum rectangle k.

1, 0, 2, 0, 1, 2, 0, 0, 2, 3, 0, 0, 1, 4, 2, 0, 0, 0, 5, 2, 4, 0, 0, 0, 3, 4, 6, 2, 0, 0, 0, 1, 4, 11, 2, 4, 0, 0, 0, 0, 3, 14, 4, 6, 3, 0, 0, 0, 0, 1, 15, 6, 12, 4, 4, 0, 0, 0, 0, 0, 13, 8, 18, 9, 6, 2, 0, 0, 0, 0, 0, 8, 10, 25, 14, 12, 2, 6, 0, 0, 0, 0, 0, 4, 9, 30, 22, 20, 4, 10, 2

Offset: 1

Franklin T. Adams-Watters, Mar 11 2006

T(n,k)=0 if n > A006218(k).

			The table starts:
  1;
  0, 2;
  0, 1, 2;
  0, 0, 2, 3;
  0, 0, 1, 4, 2;
  0, 0, 0, 5, 2,  4;
  0, 0, 0, 3, 4,  6, 2;
  0, 0, 0, 1, 4, 11, 2,  4;
  0, 0, 0, 0, 3, 14, 4,  6, 3;
  0, 0, 0, 0, 1, 15, 6, 12, 4, 4;
  ...

Alois P. Heinz, Rows n = 1..141, flattened
Eric Weisstein's World of Mathematics, Ferrers Diagram.

Cf. A000005 (diagonal), A000041 (row sums), A061017 (column indices of leftmost nonzero elements), A115724 (column sums), A115727, A115728, A006218, A182099.

Sum_{k=1..n} k * T(n,k) = A182099(n).

A115725 Number of partitions with maximum rectangle <= n.

Original entry on oeis.org

1, 2, 5, 10, 26, 42, 118, 171, 389, 692, 1442, 1854, 5534, 6895, 11910, 21116, 44278, 52568, 118734, 138670, 300326, 492507, 728514, 829244, 2167430, 2987124, 4167602, 6092588, 11308432, 12554900, 29925267, 33023589, 57950313, 81424281, 106214784, 148101088

Offset: 0

Franklin T. Adams-Watters, Mar 11 2006

nonn

A partition has maximum rectangle <= n iff it is a subpartition of row n of A010766.

			The 10 partitions with maximum rectangle <= 3: 0: []; 1: [1]; 2: [2], [1^2], [2,1]; 3: [3], [1^3], [3,1], [2,1^2], [3,1^2].

Alois P. Heinz, Table of n, a(n) for n = 0..100
Eric Weisstein's World of Mathematics, Ferrers Diagram.

Cf. A115728, A115729, A115724, A010766.

a(n) = subpart([A115728 (or A115729), [] is row n of A010766.

a(n) = Sum_{k>=0} A182114(k,n). - Alois P. Heinz, Nov 02 2012

A115726 Maximum rectangle of partitions in Abramowitz and Stegun order.

Original entry on oeis.org

0, 1, 2, 2, 3, 2, 3, 4, 3, 4, 3, 4, 5, 4, 4, 3, 4, 4, 5, 6, 5, 4, 6, 4, 4, 6, 4, 4, 5, 6, 7, 6, 5, 6, 5, 4, 6, 6, 4, 4, 6, 5, 5, 6, 7, 8, 7, 6, 6, 8, 6, 5, 6, 6, 6, 5, 4, 6, 6, 8, 5, 5, 6, 6, 6, 7, 8, 9, 8, 7, 6, 8, 7, 6, 6, 8, 6, 6, 9, 6, 5, 6, 6, 6, 8, 5, 5, 6, 6, 8, 6, 6, 6, 7, 7, 8, 9

Offset: 0

Franklin T. Adams-Watters, Mar 11 2006

nonn

			Partition 33 in A&S order is [4,3]; its maximum rectangle is 2x3 = 6, so a(33)=6.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Eric Weisstein's World of Mathematics, Ferrers Diagram.

Cf. A115726, A115724, A036036.

A115727 Maximum rectangle of partitions in Mathematica order.

Original entry on oeis.org

0, 1, 2, 2, 3, 2, 3, 4, 3, 4, 3, 4, 5, 4, 4, 3, 4, 4, 5, 6, 5, 4, 4, 6, 4, 4, 6, 4, 5, 6, 7, 6, 5, 5, 6, 4, 4, 6, 6, 4, 5, 6, 5, 6, 7, 8, 7, 6, 6, 6, 5, 5, 8, 6, 6, 4, 5, 6, 6, 6, 5, 6, 8, 6, 6, 7, 8, 9, 8, 7, 7, 6, 6, 6, 8, 6, 6, 5, 5, 8, 6, 6, 6, 5, 6, 9, 6, 6, 8, 6, 6, 7, 8, 6, 7, 8, 9

Offset: 0

Franklin T. Adams-Watters, Mar 11 2006

nonn

			Partition 34 in Mathematica order is [4,3]; its maximum rectangle is 2x3 = 6, so a(34)=6.

Eric Weisstein's World of Mathematics, Ferrers Diagram.

Cf. A115726, A115724, A080577.

cp's OEIS Frontend

A115723 Table of partitions of n with maximum rectangle k.

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A115725 Number of partitions with maximum rectangle <= n.

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A115726 Maximum rectangle of partitions in Abramowitz and Stegun order.

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A115727 Maximum rectangle of partitions in Mathematica order.

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