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%I A343535 #25 Apr 26 2021 08:53:52
%S A343535 1,1,2,5,1,20,4,102,18,626,92,2,4458,564,18,36144,4032,144,328794,
%T A343535 32898,1182,6,3316944,301248,10512,96,36755520,3057840,102240,1200,
%U A343535 443828184,34073184,1085904,14304,24,5800823880,413484240,12538080,174000,600,81591320880
%N A343535 Number T(n,k) of permutations of [n] having exactly k consecutive triples j, j+1, j-1; triangle T(n,k), n>=0, 0<=k<=floor(n/3), read by rows.
%C A343535 Terms in column k are multiples of k!.
%H A343535 Anders Claesson, <a href="https://akc.is/papers/036-From-Hertzsprungs-problem-to-pattern-rewriting-systems.pdf">From Hertzsprung's problem to pattern-rewriting systems</a>, University of Iceland (2020).
%H A343535 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>
%F A343535 T(3n,n) = n!.
%e A343535 T(4,1) = 4: 1342, 2314, 3421, 4231.
%e A343535 Triangle T(n,k) begins:
%e A343535               1;
%e A343535               1;
%e A343535               2;
%e A343535               5,           1;
%e A343535              20,           4;
%e A343535             102,          18;
%e A343535             626,          92,          2;
%e A343535            4458,         564,         18;
%e A343535           36144,        4032,        144;
%e A343535          328794,       32898,       1182,        6;
%e A343535         3316944,      301248,      10512,       96;
%e A343535        36755520,     3057840,     102240,     1200;
%e A343535       443828184,    34073184,    1085904,    14304,     24;
%e A343535      5800823880,   413484240,   12538080,   174000,    600;
%e A343535     81591320880,  5428157760,  156587040,  2214720,  10800;
%e A343535   1228888215960, 76651163160, 2105035440, 29777520, 175800, 120;
%e A343535   ...
%p A343535 b:= proc(s, l, t) option remember; `if`(s={}, 1, add((h->
%p A343535       expand(b(s minus {j}, j, `if`(h=1, 2, 1))*
%p A343535      `if`(t=2 and h=-2, x, 1)))(j-l), j=s))
%p A343535     end:
%p A343535 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(
%p A343535                b({$1..n}, -1, 1)):
%p A343535 seq(T(n), n=0..13);
%t A343535 b[s_, l_, t_] := b[s, l, t] = If[s == {}, 1, Sum[Function[h,
%t A343535      Expand[b[s ~Complement~ {j}, j, If[h == 1, 2, 1]]*
%t A343535      If[t == 2 && h == -2, x, 1]]][j - l], {j, s}]];
%t A343535 T[n_] := CoefficientList[b[Range[n], -1, 1], x];
%t A343535 T /@ Range[0, 13] // Flatten (* _Jean-François Alcover_, Apr 26 2021, after _Alois P. Heinz_ *)
%Y A343535 Column k=0 gives A212580.
%Y A343535 Row sums give A000142.
%Y A343535 Cf. A047921, A123513, A197365, A216716.
%K A343535 nonn,tabf
%O A343535 0,3
%A A343535 _Alois P. Heinz_, Apr 18 2021

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