A388712 Triangle read by rows where T(n,k) is the number of integer partitions of n into k parts without choosable initial intervals.

0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 2, 1, 1, 0, 0, 0, 2, 2, 1, 1, 0, 0, 0, 1, 3, 2, 1, 1, 0, 0, 0, 1, 5, 3, 2, 1, 1, 0, 0, 0, 1, 6, 5, 3, 2, 1, 1, 0, 0, 0, 1, 8, 7, 5, 3, 2, 1, 1, 0, 0, 0, 1, 7, 10, 7, 5, 3, 2, 1, 1

Offset

0,19

Comments

We say that a set or sequence of nonempty sets is choosable iff it is possible to choose a different element from each set. For example, ({1,2},{1},{1,3}) is choosable because we have the choice (2,1,3), but ({1},{1,2},{2}) is not. The initial interval of a nonnegative integer x is the set {1,...,x}.

Conjecture: Also the number of non-superdiagonal reversed partitions of n into k parts, where a sequence (y_1, ..., y_k) is superdiagonal iff y_i >= i for all i = 1..k.

Links

Table of n, a(n) for n=0..77.

Example

Row n = 9 counts the following partitions:
  (711)  (3222)  (22221)  (222111)  (2211111)  (21111111)  (111111111)
         (3321)  (32211)  (321111)  (3111111)
         (4221)  (33111)  (411111)
         (4311)  (42111)
         (5211)  (51111)
         (6111)
Triangle begins:
  0
  0  0
  0  0  1
  0  0  0  1
  0  0  0  1  1
  0  0  0  2  1  1
  0  0  0  2  2  1  1
  0  0  0  1  3  2  1  1
  0  0  0  1  5  3  2  1  1
  0  0  0  1  6  5  3  2  1  1
  0  0  0  1  8  7  5  3  2  1  1
  0  0  0  1  7 10  7  5  3  2  1  1
  0  0  0  1  8 13 11  7  5  3  2  1  1
  0  0  0  1  7 18 14 11  7  5  3  2  1  1
  0  0  0  1  8 23 20 15 11  7  5  3  2  1  1
  0  0  0  1  8 29 26 21 15 11  7  5  3  2  1  1
  0  0  0  1  9 32 35 28 22 15 11  7  5  3  2  1  1
  0  0  0  1  9 36 44 38 29 22 15 11  7  5  3  2  1  1
  0  0  0  1 10 38 58 49 40 30 22 15 11  7  5  3  2  1  1
  0  0  0  1 10 41 71 65 52 41 30 22 15 11  7  5  3  2  1  1
  0  0  0  1 11 45 90 82 70 54 42 30 22 15 11  7  5  3  2  1  1

Mathematica

Table[Length[Select[IntegerPartitions[n, {k}], Select[Tuples[Range/@#], UnsameQ@@#&]=={}&]], {n, 0, 10}, {k, 0, n}]

Crossrefs

For strict integer partitions the complement is A382864 or A008289.

Row sums appear to be A387118, complement A238873.

The complement is counted by A388711.

A000041 counts integer partitions, strict A000009.

A001522 counts partitions with a diagonal, complement A064428.

A003114 counts superdiagonal partitions, strictly A003106.

A114088 counts partitions by excedances.

A238394 counts strictly superdiagonal reversed partitions, complement A238395.

A238875 counts subdiagonal partitions, ranks apparently A387112.

A370592 counts choosable partitions (ranks A368100), complement A370593 (ranks A355529).

Cf. A000700, A052337, A059607, A060016, A387134, A387328, A387576, A387577, A387578, A387880, A388710, A388724.

Sequence in context: A351357 A263250 A209287 * A025901 A204431 A386244

Adjacent sequences: A388709 A388710 A388711 * A388713 A388714 A388715

Keyword

nonn,tabl

Author

Gus Wiseman, Sep 24 2025

Status

approved