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%I A268193 #27 Jul 13 2025 11:06:46
%S A268193 1,2,2,1,4,1,4,3,8,2,1,8,6,1,13,7,2,15,11,4,22,15,4,1,24,24,7,1,37,26,
%T A268193 12,2,40,42,16,3,57,50,22,6,64,72,33,6,1,89,84,46,11,1,98,122,60,15,2,
%U A268193 135,141,82,24,3,149,198,106,32,5,199,231,144,45,8,224,309,187,61,10,1
%N A268193 Triangle read by rows: T(n,k) (n>=1, k>=0) is the number of partitions of n which have k distinct parts i such that i+1 is also a part.
%C A268193 T(n,k) = number of partitions of n having k singleton parts other than the largest part. Example: T(5,1) = 3 because we have [4,1'], [3,2'], [2,2,1'] (the counted singletons are marked). These partitions are connected by conjugation to those in the definition.
%C A268193 From _Gus Wiseman_, Jul 10 2025: (Start)
%C A268193 Also the number of integer partitions of n with k maximal subsequences of consecutive parts not decreasing by 1 (anti-runs). For example, row n = 8 counts partitions with the following anti-runs:
%C A268193 ((8)) ((3,3),(2)) ((3),(2,2),(1))
%C A268193 ((4,4)) ((4),(3,1)) ((3),(2),(1,1,1))
%C A268193 ((5,3)) ((5,2),(1))
%C A268193 ((6,2)) ((4,2),(1,1))
%C A268193 ((7,1)) ((2,2,2),(1,1))
%C A268193 ((4,2,2)) ((2,2),(1,1,1,1))
%C A268193 ((6,1,1)) ((2),(1,1,1,1,1,1))
%C A268193 ((2,2,2,2))
%C A268193 ((3,3,1,1))
%C A268193 ((5,1,1,1))
%C A268193 ((4,1,1,1,1))
%C A268193 ((3,1,1,1,1,1))
%C A268193 ((1,1,1,1,1,1,1,1))
%C A268193 (End)
%H A268193 Alois P. Heinz, <a href="/A268193/b268193.txt">Rows n = 1..800, flattened</a>
%F A268193 T(n,0) = A116931(n).
%F A268193 Sum_{k>=1} T(n, k) = A000041(n) (the partition numbers).
%F A268193 Sum_{k>=1} k*T(n,k) = A024786(n-1).
%F A268193 G.f.: G(t,x) = Sum_{j>=1} ((x^j/(1-x^j))*Product_{i=1..j-1} (1 + tx^i + x^{2i}/(1-x^i))).
%e A268193 T(5,1) = 3 because we have [3,2], [2,2,1], and [2,1,1,1].
%e A268193 T(9,2) = 4 because we have [3,2',1,1,1,1'], [3,2,2',1,1'], [3,3,2',1'], and [4,3',2'] (the i's are marked).
%e A268193 Triangle starts:
%e A268193 1;
%e A268193 2;
%e A268193 2,1;
%e A268193 4,1;
%e A268193 4,3;
%e A268193 8,2,1;
%e A268193 8,6,1;
%e A268193 From _Gus Wiseman_, Jul 11 2025: (Start)
%e A268193 Row n = 8 counts the following partitions by number of singleton parts other than the largest part:
%e A268193 (8) (5,3) (4,3,1)
%e A268193 (4,4) (6,2) (5,2,1)
%e A268193 (4,2,2) (7,1)
%e A268193 (6,1,1) (3,3,2)
%e A268193 (2,2,2,2) (3,2,2,1)
%e A268193 (3,3,1,1) (4,2,1,1)
%e A268193 (5,1,1,1) (3,2,1,1,1)
%e A268193 (2,2,2,1,1)
%e A268193 (4,1,1,1,1)
%e A268193 (2,2,1,1,1,1)
%e A268193 (3,1,1,1,1,1)
%e A268193 (2,1,1,1,1,1,1)
%e A268193 (1,1,1,1,1,1,1,1)
%e A268193 (End)
%p A268193 g := add(x^j*mul(1+t*x^i+x^(2*i)/(1-x^i), i = 1 .. j-1)/(1-x^j), j = 1 .. 80): gser := simplify(series(g, x = 0, 27)): for n from 0 to 25 do P[n] := sort(coeff(gser, x, n)) end do: for n to 25 do seq(coeff(P[n], t, k), k = 0 .. degree(P[n])) end do; # yields sequence in triangular form
%p A268193 # second Maple program:
%p A268193 b:= proc(n, i, t) option remember; expand(`if`(n=0, 1,
%p A268193 `if`(i<1, 0, add(b(n-i*j, i-1, t or j>0)*
%p A268193 `if`(t and j=1, x, 1), j=0..n/i))))
%p A268193 end:
%p A268193 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2, false)):
%p A268193 seq(T(n), n=1..20); # _Alois P. Heinz_, Feb 13 2016
%t A268193 b[n_, i_, t_] := b[n, i, t] = Expand[If[n == 0, 1, If[i < 1, 0, Sum[b[n - i*j, i - 1, t || j > 0]*If[t && j == 1, x, 1], {j, 0, n/i}]]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, n, False]]; Table[T[n], {n, 1, 20}] // Flatten (* _Jean-François Alcover_, Dec 21 2016, after _Alois P. Heinz_ *)
%t A268193 Table[Length[Select[IntegerPartitions[n],Length[Split[#,#1!=#2+1&]]==k&]],{n,0,10},{k,0,n}] (* Delete zeros for A268193. _Gus Wiseman_, Jul 10 2025 *)
%Y A268193 Row sums are A000041.
%Y A268193 Row lengths are A003056.
%Y A268193 For distinct parts instead of anti-runs we have A116608.
%Y A268193 Column k = 1 is A116931.
%Y A268193 For runs instead of anti-runs we have A384881.
%Y A268193 The strict case is A384905.
%Y A268193 The corresponding rank statistic is A356228, non-strict version A384906.
%Y A268193 The proper case is A385814, runs A385815.
%Y A268193 A007690 counts partitions with no singletons, complement A183558.
%Y A268193 A034296 counts flat or gapless partitions, ranks A066311 or A073491.
%Y A268193 Cf. A000009, A008284, A024786, A047993, A098859, A116674, A287170, A325325, A384882, A385213.
%K A268193 nonn,look,tabf
%O A268193 1,2
%A A268193 _Emeric Deutsch_, Feb 13 2016