A388710 Triangle read by rows where T(n,k) is the number of integer partitions of n into k parts, none of which is less than its own multiplicity.

1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 2, 0, 0, 0, 1, 2, 1, 0, 0, 0, 1, 3, 1, 0, 0, 0, 0, 1, 3, 3, 0, 0, 0, 0, 0, 1, 4, 4, 1, 0, 0, 0, 0, 0, 1, 4, 6, 2, 0, 0, 0, 0, 0, 0, 1, 5, 7, 4, 0, 0, 0, 0, 0, 0, 0, 1, 5, 9, 6, 1, 0, 0, 0, 0, 0, 0, 0, 1, 6, 11, 8, 2, 0, 0, 0, 0, 0, 0, 0, 0, 1, 6, 13, 12, 4, 0, 0, 0, 0, 0, 0, 0, 0

Offset

0,13

Comments

These are partitions with at most x x's for every x.

Links

Alois P. Heinz, Rows n = 0..200, flattened

Example

Row n = 9 counts the following partitions:
  .  (9)  (81)  (621)  (4221)  .  .  .  .  .
          (72)  (531)  (3321)
          (63)  (522)
          (54)  (441)
                (432)
                (333)
Triangle begins:
  1
  0  1
  0  1  0
  0  1  1  0
  0  1  2  0  0
  0  1  2  1  0  0
  0  1  3  1  0  0  0
  0  1  3  3  0  0  0  0
  0  1  4  4  1  0  0  0  0
  0  1  4  6  2  0  0  0  0  0
  0  1  5  7  4  0  0  0  0  0  0
  0  1  5  9  6  1  0  0  0  0  0  0
  0  1  6 11  8  2  0  0  0  0  0  0  0
  0  1  6 13 12  4  0  0  0  0  0  0  0  0
  0  1  7 15 16  6  1  0  0  0  0  0  0  0  0
  0  1  7 18 20 11  1  0  0  0  0  0  0  0  0  0
  0  1  8 20 26 15  3  0  0  0  0  0  0  0  0  0  0
  0  1  8 23 31 22  6  0  0  0  0  0  0  0  0  0  0  0
  0  1  9 26 38 29 10  1  0  0  0  0  0  0  0  0  0  0  0
  0  1  9 29 45 39 16  2  0  0  0  0  0  0  0  0  0  0  0  0

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
     b(n, i-1)+add(expand(x^j*b(n-i*j, i-1)), j=1..min(i, n/i))))
    end:
T:= (n, k)-> coeff(b(n$2), x, k):
seq(seq(T(n, k), k=0..n), n=0..14);  # Alois P. Heinz, Oct 03 2025

Mathematica

Table[Length[Select[IntegerPartitions[n, {k}], And@@Table[Count[#, k]<=k, {k, Union[#]}]&]], {n, 0, 10}, {k, 0, n}]

Crossrefs

For constant partitions see A038548.

Row sums are A052335, ranks A276078, A276079.

For strict integer partitions we have A060016.

For strictly superdiagonal see A087153, ranks A325128.

A000041 counts integer partitions, strict A000009.

A001522 counts partitions with a diagonal, complement A064428.

A003106 counts strictly superdiagonal partitions, strict A237979, ranks A352830.

A003114 counts superdiagonal partitions, strict partitions A025157.

A114088 counts partitions by excedances.

A238394 counts strictly superdiagonal reversed partitions, complement A238395.

A238873 counts superdiagonal reversed partitions (subdiagonal A238875), ranks A387112.

A352833 counts partitions by fixed points, reversed partitions A238352.

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

Cf. A052337, A059607, A120383, A239312, A335433, A352828, A387576, A387578, A387880, A388711, A388724.

Sequence in context: A218787 A325336 A060016 * A117408 A228360 A303138

Adjacent sequences: A388707 A388708 A388709 * A388711 A388712 A388713

Keyword

nonn,tabl

Author

Gus Wiseman, Sep 23 2025

Status

approved