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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 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-François 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.)