A343348 Irregular triangle read by rows where T(n,k) is the number of strict integer partitions of n with least gap k.

1, 0, 1, 1, 0, 1, 0, 1, 1, 1, 0, 2, 1, 0, 2, 1, 0, 1, 3, 1, 1, 0, 3, 2, 1, 0, 5, 2, 1, 0, 5, 3, 1, 0, 1, 7, 3, 1, 1, 0, 8, 4, 2, 1, 0, 10, 5, 2, 1, 0, 12, 6, 3, 1, 0, 15, 7, 3, 1, 0, 1, 17, 9, 4, 1, 1, 0, 21, 10, 4, 2, 1, 0, 25, 12, 6, 2, 1, 0, 29, 15, 6, 3, 1, 0, 35, 17, 8, 3, 1, 0

Offset: 0

Author

Gus Wiseman, Apr 18 2021

Keywords

• nonn
• tabf

Comments

The least gap (or mex) of a partition is the least positive integer that is not a part.

Row lengths are chosen to be consistent with the non-strict case A264401.

Examples

			Triangle begins:
   1
   0   1
   1   0
   1   0   1
   1   1   0
   2   1   0
   2   1   0   1
   3   1   1   0
   3   2   1   0
   5   2   1   0
   5   3   1   0   1
   7   3   1   1   0
   8   4   2   1   0
  10   5   2   1   0
  12   6   3   1   0
  15   7   3   1   0   1
		

Links

• George E. Andrews and David Newman, Partitions and the Minimal Excludant, Annals of Combinatorics, Volume 23, May 2019, Pages 249-254.
• Brian Hopkins, James A. Sellers, and Dennis Stanton, Dyson's Crank and the Mex of Integer Partitions, arXiv:2009.10873 [math.CO], 2020.

Crossrefs

Row sums are A000009.

Row lengths are A002024.

Column k = 1 is A025147.

Column k = 2 is A025148.

The non-strict version is A264401.

A000009 counts strict partitions.

A000041 counts partitions.

A000070 counts partitions with a selected part.

A006128 counts partitions with a selected position.

A015723 counts strict partitions with a selected part.

A257993 gives the least gap of the partition with Heinz number n.

A339564 counts factorizations with a selected factor.

A342050 ranks partitions with even least gap.

A342051 ranks partitions with odd least gap.

Cf. A003242, A083710, A083711, A097986, A098743, A098965, A130689, A200745, A341450, A343347, A343377.

Programs

Mathematica

mingap[q_]:=Min@@Complement[Range[If[q=={},0,Max[q]]+1],q];
Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&mingap[#]==k&]],{n,0,15},{k,Round[Sqrt[2*(n+1)]]}]