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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.

%I A343348 #8 Apr 19 2021 15:00:55
%S A343348 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 A343348 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 A343348 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 A343348 Irregular triangle read by rows where T(n,k) is the number of strict integer partitions of n with least gap k.
%C A343348 The least gap (or mex) of a partition is the least positive integer that is not a part.
%C A343348 Row lengths are chosen to be consistent with the non-strict case A264401.
%H A343348 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 A343348 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 A343348 Triangle begins:
%e A343348    1
%e A343348    0   1
%e A343348    1   0
%e A343348    1   0   1
%e A343348    1   1   0
%e A343348    2   1   0
%e A343348    2   1   0   1
%e A343348    3   1   1   0
%e A343348    3   2   1   0
%e A343348    5   2   1   0
%e A343348    5   3   1   0   1
%e A343348    7   3   1   1   0
%e A343348    8   4   2   1   0
%e A343348   10   5   2   1   0
%e A343348   12   6   3   1   0
%e A343348   15   7   3   1   0   1
%t A343348 mingap[q_]:=Min@@Complement[Range[If[q=={},0,Max[q]]+1],q];
%t A343348 Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&mingap[#]==k&]],{n,0,15},{k,Round[Sqrt[2*(n+1)]]}]
%Y A343348 Row sums are A000009.
%Y A343348 Row lengths are A002024.
%Y A343348 Column k = 1 is A025147.
%Y A343348 Column k = 2 is A025148.
%Y A343348 The non-strict version is A264401.
%Y A343348 A000009 counts strict partitions.
%Y A343348 A000041 counts partitions.
%Y A343348 A000070 counts partitions with a selected part.
%Y A343348 A006128 counts partitions with a selected position.
%Y A343348 A015723 counts strict partitions with a selected part.
%Y A343348 A257993 gives the least gap of the partition with Heinz number n.
%Y A343348 A339564 counts factorizations with a selected factor.
%Y A343348 A342050 ranks partitions with even least gap.
%Y A343348 A342051 ranks partitions with odd least gap.
%Y A343348 Cf. A003242, A083710, A083711, A097986, A098743, A098965, A130689, A200745, A341450, A343347, A343377.
%K A343348 nonn,tabf
%O A343348 0,12
%A A343348 _Gus Wiseman_, Apr 18 2021