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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. 1
1, 1, 2, 5, 1, 20, 4, 102, 18, 626, 92, 2, 4458, 564, 18, 36144, 4032, 144, 328794, 32898, 1182, 6, 3316944, 301248, 10512, 96, 36755520, 3057840, 102240, 1200, 443828184, 34073184, 1085904, 14304, 24, 5800823880, 413484240, 12538080, 174000, 600, 81591320880 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Terms in column k are multiples of k!.

LINKS

Table of n, a(n) for n=0..40.

Anders Claesson, From Hertzsprung's problem to pattern-rewriting systems, University of Iceland (2020).

Wikipedia, Permutation

FORMULA

T(3n,n) = n!.

EXAMPLE

T(4,1) = 4: 1342, 2314, 3421, 4231.

Triangle T(n,k) begins:

              1;

              1;

              2;

              5,           1;

             20,           4;

            102,          18;

            626,          92,          2;

           4458,         564,         18;

          36144,        4032,        144;

         328794,       32898,       1182,        6;

        3316944,      301248,      10512,       96;

       36755520,     3057840,     102240,     1200;

      443828184,    34073184,    1085904,    14304,     24;

     5800823880,   413484240,   12538080,   174000,    600;

    81591320880,  5428157760,  156587040,  2214720,  10800;

  1228888215960, 76651163160, 2105035440, 29777520, 175800, 120;

  ...

MAPLE

b:= proc(s, l, t) option remember; `if`(s={}, 1, add((h->

      expand(b(s minus {j}, j, `if`(h=1, 2, 1))*

     `if`(t=2 and h=-2, x, 1)))(j-l), j=s))

    end:

T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(

               b({$1..n}, -1, 1)):

seq(T(n), n=0..13);

MATHEMATICA

b[s_, l_, t_] := b[s, l, t] = If[s == {}, 1, Sum[Function[h,

     Expand[b[s ~Complement~ {j}, j, If[h == 1, 2, 1]]*

     If[t == 2 && h == -2, x, 1]]][j - l], {j, s}]];

T[n_] := CoefficientList[b[Range[n], -1, 1], x];

T /@ Range[0, 13] // Flatten (* Jean-Fran├žois Alcover, Apr 26 2021, after Alois P. Heinz *)

CROSSREFS

Column k=0 gives A212580.

Row sums give A000142.

Cf. A047921, A123513, A197365, A216716.

Sequence in context: A187244 A120294 A186766 * A047921 A242783 A177250

Adjacent sequences:  A343532 A343533 A343534 * A343536 A343537 A343538

KEYWORD

nonn,tabf

AUTHOR

Alois P. Heinz, Apr 18 2021

STATUS

approved

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Last modified September 17 17:29 EDT 2021. Contains 347489 sequences. (Running on oeis4.)