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A264401 Triangle read by rows: T(n,k) is the number of partitions of n having least gap k.

%I A264401 #37 May 21 2021 11:29:24
%S A264401 1,0,1,1,1,1,1,1,2,2,1,2,3,2,4,4,2,1,4,6,4,1,7,8,5,2,8,11,8,3,12,15,
%T A264401 10,4,1,14,20,15,6,1,21,26,19,9,2,24,35,27,12,3,34,45,34,17,5,41,58,
%U A264401 47,23,6,1,55,75,59,31,10,1,66,96,79,41,13,2
%N A264401 Triangle read by rows: T(n,k) is the number of partitions of n having least gap k.
%C A264401 The "least gap" or "mex" of a partition is the least positive integer that is not a part of the partition. For example, the least gap of the partition [7,4,2,2,1] is 3.
%C A264401 Sum of entries in row n is A000041(n).
%C A264401 T(n,1) = A002865(n).
%C A264401 Sum_{k>=1} k*T(n,k) = A022567(n).
%H A264401 Alois P. Heinz, <a href="/A264401/b264401.txt">Rows n = 0..1000, flattened</a>
%H A264401 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 A264401 P. J. Grabner and A. Knopfmacher, <a href="http://www.math.tugraz.at/fosp/pdfs/tugraz_0087.pdf">Analysis of some new partition statistics</a>, Ramanujan J., 12, 2006, 439-454.
%H A264401 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.
%H A264401 Wikipedia, <a href="https://en.wikipedia.org/wiki/Mex_(mathematics)">Mex (mathematics)</a>
%F A264401 G.f.: G(t,x) = Sum_{j>=1} (t^j*x^{j(j-1)/2}*(1-x^j))/Product_{i>=1}(1-x^i).
%e A264401 Row n=5 is 2,3,2; indeed, the least gaps of [5], [4,1], [3,2], [3,1,1], [2,2,1], [2,1,1,1], and [1,1,1,1,1] are 1, 2, 1, 2, 3, 3, and 2, respectively (i.e., two 1s, three 2s, and two 3s).
%e A264401 Triangle begins:
%e A264401    1
%e A264401    0   1
%e A264401    1   1
%e A264401    1   1   1
%e A264401    2   2   1
%e A264401    2   3   2
%e A264401    4   4   2   1
%e A264401    4   6   4   1
%e A264401    7   8   5   2
%e A264401    8  11   8   3
%e A264401   12  15  10   4   1
%e A264401   14  20  15   6   1
%e A264401   21  26  19   9   2
%p A264401 g := (sum(t^j*x^((1/2)*j*(j-1))*(1-x^j), j = 1 .. 80))/(product(1-x^i, i = 1 .. 80)): gser := simplify(series(g, x = 0, 23)): for n from 0 to 30 do P[n] := sort(coeff(gser, x, n)) end do: for n from 0 to 25 do seq(coeff(P[n], t, j), j = 1 .. degree(P[n])) end do; # yields sequence in triangular form
%p A264401 # second Maple program:
%p A264401 b:= proc(n, i) option remember; `if`(n=0, `if`(i=0, [1, 0],
%p A264401       [0, x]), `if`(i<1, 0, (p-> [0, p[2] +p[1]*x^i])(
%p A264401       b(n, i-1)) +add(b(n-i*j, i-1), j=1..n/i)))
%p A264401     end:
%p A264401 T:= n->(p->seq(coeff(p, x, i), i=1..degree(p)))(b(n, n+1)[2]):
%p A264401 seq(T(n), n=0..20);  # _Alois P. Heinz_, Nov 29 2015
%t A264401 Needs["Combinatorica`"]; {1, 0}~Join~Flatten[Table[Count[Map[If[# == {}, 0, First@ #] &@ Complement[Range@ n, #] &, Combinatorica`Partitions@ n], n_ /; n == k], {n, 17}, {k, n}] /. 0 -> Nothing] (* _Michael De Vlieger_, Nov 21 2015 *)
%t A264401 mingap[q_]:=Min@@Complement[Range[If[q=={},0,Max[q]]+1],q];Table[Length[Select[IntegerPartitions[n],mingap[#]==k&]],{n,0,15},{k,Round[Sqrt[2*(n+1)]]}] (* _Gus Wiseman_, Apr 19 2021 *)
%t A264401 b[n_, i_] := b[n, i] = If[n == 0, If[i == 0, {1, 0}, {0, x}], If[i<1, {0, 0}, {0, #[[2]] + #[[1]]*x^i}&[b[n, i-1]] + Sum[b[n-i*j, i - 1], {j, 1, n/i}]]];
%t A264401 T[n_] := CoefficientList[b[n, n + 1], x][[2]] // Rest;
%t A264401 T /@ Range[0, 20] // Flatten (* _Jean-François Alcover_, May 21 2021, after _Alois P. Heinz_ *)
%Y A264401 Row sums are A000041.
%Y A264401 Row lengths are A002024.
%Y A264401 Column k = 1 is A002865.
%Y A264401 Column k = 2 is A027336.
%Y A264401 The strict case is A343348.
%Y A264401 A000009 counts strict partitions.
%Y A264401 A000041 counts partitions.
%Y A264401 A000070 counts partitions with a selected part.
%Y A264401 A006128 counts partitions with a selected position.
%Y A264401 A015723 counts strict partitions with a selected part.
%Y A264401 A257993 gives the least gap of the partition with Heinz number n.
%Y A264401 A339564 counts factorizations with a selected factor.
%Y A264401 A342050 ranks partitions with even least gap.
%Y A264401 A342051 ranks partitions with odd least gap.
%Y A264401 Cf. A003242, A022567, A048004, A083710, A098743, A130689, A338470, A343341.
%K A264401 nonn,tabf
%O A264401 0,9
%A A264401 _Emeric Deutsch_, Nov 21 2015