A115727 -id:A115727 - cp's OEIS frontend

A115724 Number of partitions with maximum rectangle n.

1, 1, 3, 5, 16, 16, 76, 53, 218, 303, 750, 412, 3680, 1361, 5015, 9206, 23162, 8290, 66166, 19936, 161656, 192181, 236007, 100730, 1338186, 819694, 1180478, 1924986, 5215844, 1246468, 17370367, 3098322, 24926724, 23473968, 24790503, 41886304, 227243488

Offset: 0

Franklin T. Adams-Watters, Mar 11 2006

nonn

			The 16 partitions with maximum rectangle 4 are [4], [2^2], [1^4], [4,1], [3,2], [2^2,1], [2,1^3], [4,2], [4,1^2], [3,2,1], [3,1^3], [2^2,1^2], [4,2,1], [4,1^3], [3,2,1^2] and [4,2,1^2].

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

Cf. A115725, A115723, A115726, A115727.

a(n) = A115725(n) - A115725(n-1).

A115723 Table of partitions of n with maximum rectangle k.

Original entry on oeis.org

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).

A115628 Factorization of n! into primorials.

Original entry on oeis.org

1, 0, 1, 2, 1, 2, 0, 1, 2, 1, 1, 2, 1, 0, 1, 5, 1, 0, 1, 3, 3, 0, 1, 4, 2, 1, 1, 4, 2, 1, 0, 1, 5, 3, 1, 0, 1, 5, 3, 1, 0, 0, 1, 6, 3, 0, 1, 0, 1, 5, 3, 1, 1, 0, 1, 9, 3, 1, 1, 0, 1, 9, 3, 1, 1, 0, 0, 1, 8, 5, 1, 1, 0, 0, 1, 8, 5, 1, 1, 0, 0, 0, 1, 10, 4, 2, 1, 0, 0, 0, 1, 9, 5, 1, 2, 0, 0, 0, 1, 10, 5, 1, 1, 1

Offset: 2

Franklin T. Adams-Watters, Jan 26 2006

Factorial of n has a unique representation as a product of primorials: n! = 2#^T(n,1)*3#^T(n,2)*...*P_{Pi(n)}#^T(n,Pi(n)).

			Rows start: 1; 0,1; 2,1; 2,0,1; 2,1,1; 2,1,0,1; 5,1,0,1; 3,3,0,1.

Cf. A115627, A000142, A002110. Row lengths are A000720.

T(n, k) = A115727(n, k)-A115727(n, k+1).

cp's OEIS Frontend

A115724 Number of partitions with maximum rectangle n.

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A115723 Table of partitions of n with maximum rectangle k.

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A115628 Factorization of n! into primorials.

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