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A343348
Irregular triangle read by rows where T(n,k) is the number of strict integer partitions of n with least gap k.
2
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
0,12
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.
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.
EXAMPLE
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
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)]]}]
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.
Sequence in context: A287072 A095734 A137269 * A112201 A112203 A196279
KEYWORD
nonn,tabf
AUTHOR
Gus Wiseman, Apr 18 2021
STATUS
approved