A393310 Triangle read by rows: T(n,k) is the number of chains of nonempty partitions ordered by inclusion that end with a partition of n and have length k.

1, 2, 2, 3, 7, 4, 5, 20, 25, 10, 7, 44, 93, 82, 26, 11, 98, 303, 432, 292, 76, 15, 189, 821, 1711, 1872, 1040, 232, 22, 369, 2125, 6005, 9389, 8314, 3916, 764, 30, 670, 5024, 18590, 39086, 49158, 36728, 15070, 2620, 42, 1212, 11521, 54085, 146473, 244889, 257636, 166546, 60556, 9496

Offset

1,2

Comments

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

Links

Table of n, a(n) for n=1..55.

John Tyler Rascoe, Python code.

Example

Triangle begins:
      k= 0      1      2      3       4       5       6       7      8     9
  n= 1  [1]
  n= 2  [2,     2]
  n= 3  [3,     7,     4]
  n= 4  [5,    20,    25,    10]
  n= 5  [7,    44,    93,    82,     26]
  n= 6  [11,   98,   303,   432,    292,     76]
  n= 7  [15,  189,   821,  1711,   1872,   1040,    232]
  n= 8  [22,  369,  2125,  6005,   9389,   8314,   3916,    764]
  n= 9  [30,  670,  5024, 18590,  39086,  49158,  36728,  15070,  2620]
  n=10  [42, 1212, 11521, 54085, 146473, 244889, 257636, 166546, 60556, 9496]
  ...
The chain of partitions (2)<(21)<(211)<(421) has length 3 and the last element is a partition of 7 so it is counted under T(7,3) = 1711.
T(3,1) = 7 counts: (1)<(111), (11)<(111), (1)<(21), (2)<(21), (11)<(21), (1)<(3), (2)<(3).

Prog

(Python) # see links

Crossrefs

Cf. A000041 (column k=0), A000085 (main diagonal), A299926 (row sums), A389604, A392995.

Sequence in context: A210753 A208762 A209746 * A354950 A267822 A210598

Adjacent sequences: A393307 A393308 A393309 * A393311 A393312 A393313

Keyword

nonn,tabl

Author

John Tyler Rascoe, Feb 10 2026

Status

approved