A342531 Triangle read by rows where T(n,k) is the number of strict integer partitions of n with maximal descent k, n >= 0, 0 <= k <= n.

1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 2, 1, 1, 0, 1, 0, 0, 1, 2, 1, 1, 1, 1, 0, 1, 0, 0, 1, 1, 2, 2, 1, 1, 1, 0, 1, 0, 0, 1, 1, 2, 3, 1, 1, 1, 1, 0, 1, 0, 0

Offset

0,39

Comments

The maximal descent of an empty or singleton partition is considered to be 0.

Links

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

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

Example

Triangle begins:
1
1 0
1 0 0
1 1 0 0
1 0 1 0 0
1 1 0 1 0 0
1 1 1 0 1 0 0
1 1 1 1 0 1 0 0
1 0 2 1 1 0 1 0 0
1 2 1 1 1 1 0 1 0 0
1 1 2 2 1 1 1 0 1 0 0
1 1 2 3 1 1 1 1 0 1 0 0
1 1 3 2 3 1 1 1 1 0 1 0 0
1 1 3 3 3 2 1 1 1 1 0 1 0 0
1 1 3 4 3 3 2 1 1 1 1 0 1 0 0
1 3 3 4 4 3 2 2 1 1 1 1 0 1 0 0
1 0 5 5 5 4 3 2 2 1 1 1 1 0 1 0 0
1 1 4 7 5 5 4 2 2 2 1 1 1 1 0 1 0 0
1 2 5 6 7 6 4 4 2 2 2 1 1 1 1 0 1 0 0
1 1 5 9 7 7 6 4 3 2 2 2 1 1 1 1 0 1 0 0
1 1 6 9 9 7 8 5 4 3 2 2 2 1 1 1 1 0 1 0 0
Row n = 15 counts the following strict partitions (empty columns indicated by dots, A..F = 10..15):
  F  87     753   96    762   A5   A41   B4   B31  C3  C21  D2  .  E1  .  .
     654    6432  852   843   861  9321  A32
     54321  6531  7431  951   942
                  7521  8421

Mathematica

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&If[Length[#]<=1, k==0, Max[Differences[Reverse[#]]]==k]&]], {n, 0, 15}, {k, 0, n}]

Crossrefs

The non-strict version is A238353.

A000041 counts partitions (strict: A000009).

A049980 counts strict partitions with equal differences.

A325325 counts partitions with distinct differences (ranking: A325368).

A325545 counts compositions with distinct differences.

Cf. A003114, A003242, A005117, A034296, A049988, A238710, A342098, A342514.

Sequence in context: A145679 A007273 A016319 * A287156 A092510 A117208

Adjacent sequences: A342528 A342529 A342530 * A342532 A342533 A342534

Keyword

nonn,tabl

Author

Gus Wiseman, Mar 25 2021

Status

approved