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A343348
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Irregular triangle read by rows where T(n,k) is the number of strict integer partitions of n with least gap k.
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2
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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
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OFFSET
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0,12
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COMMENTS
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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.
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LINKS
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EXAMPLE
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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
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MATHEMATICA
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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)]]}]
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CROSSREFS
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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.
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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