A338621 -id:A338621 - cp's OEIS frontend

A096771 Triangle read by rows: T(n,m) is the number of partitions of n that (just) fit inside an m X m box, but not in an (m-1) X (m-1) box. Partitions of n with Max(max part, length) = m.

1, 0, 2, 0, 1, 2, 0, 1, 2, 2, 0, 0, 3, 2, 2, 0, 0, 3, 4, 2, 2, 0, 0, 2, 5, 4, 2, 2, 0, 0, 1, 7, 6, 4, 2, 2, 0, 0, 1, 6, 9, 6, 4, 2, 2, 0, 0, 0, 7, 11, 10, 6, 4, 2, 2, 0, 0, 0, 5, 14, 13, 10, 6, 4, 2, 2, 0, 0, 0, 5, 15, 19, 14, 10, 6, 4, 2, 2, 0, 0, 0, 3, 17, 22, 21, 14, 10, 6, 4, 2, 2, 0, 0, 0, 2, 17, 29

Offset: 1

Wouter Meeussen, Aug 21 2004

Row sums are A000041. Columns are finite and sum to A051924. The final floor(n/2) terms of each row are the reverse of the initial terms of 2*A000041.

			T(5,3)=3, counting 32, 311 and 221.
From _Gus Wiseman_, Apr 12 2019: (Start)
Triangle begins:
  1
  0  2
  0  1  2
  0  1  2  2
  0  0  3  2  2
  0  0  3  4  2  2
  0  0  2  5  4  2  2
  0  0  1  7  6  4  2  2
  0  0  1  6  9  6  4  2  2
  0  0  0  7 11 10  6  4  2  2
  0  0  0  5 14 13 10  6  4  2  2
  0  0  0  5 15 19 14 10  6  4  2  2
  0  0  0  3 17 22 21 14 10  6  4  2  2
  0  0  0  2 17 29 27 22 14 10  6  4  2  2
  0  0  0  1 17 33 36 29 22 14 10  6  4  2  2
  0  0  0  1 15 39 45 41 30 22 14 10  6  4  2  2
  0  0  0  0 14 41 57 52 43 30 22 14 10  6  4  2  2
  0  0  0  0 11 47 67 69 57 44 30 22 14 10  6  4  2  2
  0  0  0  0  9 46 81 85 76 59 44 30 22 14 10  6  4  2  2
(End)

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)
Findstat, FindStat - St000784: The maximum of the length and the largest part of the integer partition

A version with reflected rows is A338621.
Cf. A051924, A096272, A096597, A115994, A252464, A263297, A325193, A368985.
Related triangles are A115720, A325188, A325189, A325192, A325200, with Heinz-encoded versions A257990, A325169, A065770, A325178, A325195.

Table[Count[Partitions[n], q_List /; Max[Length[q], Max[q]]===k], {n, 16}, {k, n}]

row(n)={my(r=vector(n)); forpart(p=n, r[max(#p,p[#p])]++); r} \\ Andrew Howroyd, Jan 12 2024

Sum_{k>=1} k*T(n,k) = A368985(n). - Andrew Howroyd, Jan 12 2024

cp's OEIS Frontend

A096771 Triangle read by rows: T(n,m) is the number of partitions of n that (just) fit inside an m X m box, but not in an (m-1) X (m-1) box. Partitions of n with Max(max part, length) = m.

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