A343348 - OEIS

%I #8 Apr 19 2021 15:00:55

%S 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,

%T 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,

%U 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

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

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

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

%H George E. Andrews and David Newman, <a href="https://doi.org/10.1007/s00026-019-00427-w">Partitions and the Minimal Excludant</a>, Annals of Combinatorics, Volume 23, May 2019, Pages 249-254.

%H Brian Hopkins, James A. Sellers, and Dennis Stanton, <a href="https://arxiv.org/abs/2009.10873">Dyson's Crank and the Mex of Integer Partitions</a>, arXiv:2009.10873 [math.CO], 2020.

%e Triangle begins:

%e    1

%e    0   1

%e    1   0

%e    1   0   1

%e    1   1   0

%e    2   1   0

%e    2   1   0   1

%e    3   1   1   0

%e    3   2   1   0

%e    5   2   1   0

%e    5   3   1   0   1

%e    7   3   1   1   0

%e    8   4   2   1   0

%e   10   5   2   1   0

%e   12   6   3   1   0

%e   15   7   3   1   0   1

%t mingap[q_]:=Min@@Complement[Range[If[q=={},0,Max[q]]+1],q];

%t Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&mingap[#]==k&]],{n,0,15},{k,Round[Sqrt[2*(n+1)]]}]

%Y Row sums are A000009.

%Y Row lengths are A002024.

%Y Column k = 1 is A025147.

%Y Column k = 2 is A025148.

%Y The non-strict version is A264401.

%Y A000009 counts strict partitions.

%Y A000041 counts partitions.

%Y A000070 counts partitions with a selected part.

%Y A006128 counts partitions with a selected position.

%Y A015723 counts strict partitions with a selected part.

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

%Y A339564 counts factorizations with a selected factor.

%Y A342050 ranks partitions with even least gap.

%Y A342051 ranks partitions with odd least gap.

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

%K nonn,tabf

%O 0,12

%A _Gus Wiseman_, Apr 18 2021