A389604 Triangle read by rows: T(n,k) is the number of chains in the poset of plane partitions ordered by inclusion that end with a plane partition of n and have length k.

1, 3, 3, 6, 15, 9, 13, 58, 78, 33, 24, 180, 423, 402, 135, 48, 543, 1947, 3090, 2271, 633, 86, 1468, 7667, 18237, 22122, 13377, 3207, 160, 3900, 28197, 93715, 166137, 162966, 83676, 17589, 282, 9801, 96087, 430068, 1047081, 1483233, 1224519, 547494, 102627

Offset

1,2

Comments

The length of a chain is the number of relations, so chains of length 0 are just single plane partitions of n.

Links

Table of n, a(n) for n=1..45.

John Tyler Rascoe, Python code.

Example

Triangle begins:
        k=0     1      2       3        4        5        6       7       8
  n=1  [  1]
  n=2  [  3,    3]
  n=3  [  6,   15,     9]
  n=4  [ 13,   58,    78,     33]
  n=5  [ 24,  180,   423,    402,     135]
  n=6  [ 48,  543,  1947,   3090,    2271,     633]
  n=7  [ 86, 1468,  7667,  18237,   22122,   13377,    3207]
  n=8  [160, 3900, 28197,  93715,  166137,  162966,   83676,  17589]
  n=9  [282, 9801, 96087, 430068, 1047081, 1483233, 1224519, 547494, 102627]
  ...
The chain of plane partitions below has length 2 and the last element is a plane partition of 9 so it is counted under T(9,2) = 96087.
  3 1  <  3 2 1  <  4 2 1.
  1       1         2
T(2,1) = 3 counts:
  1  <  1 1,  1  <  1   1  <  2.
                    1,

Prog

(Python) # see links

Crossrefs

Cf. A000219 (column k=0), A207542 (main diagonal), A379277, A385373, A389605 (row sums).

Sequence in context: A145597 A143418 A336452 * A092370 A245796 A345908

Adjacent sequences: A389601 A389602 A389603 * A389605 A389606 A389607

Keyword

nonn,tabl,hard

Author

John Tyler Rascoe, Oct 08 2025

Status

approved