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A323671 Number T(n,k) of permutations p of [n] with no fixed points such that |{ j : |p(j)-j| = 1 }| = k; triangle T(n,k), n >= 0, 0 <= k <= n, read by rows. 5
1, 0, 0, 0, 0, 1, 0, 0, 2, 0, 1, 2, 3, 2, 1, 4, 12, 14, 8, 6, 0, 29, 68, 82, 54, 25, 6, 1, 206, 496, 546, 376, 170, 48, 12, 0, 1708, 3960, 4349, 2922, 1353, 430, 98, 12, 1, 15702, 35816, 38632, 26048, 12084, 4052, 982, 160, 20, 0, 159737, 358786, 383523, 257552, 120919, 41508, 10647, 1998, 270, 20, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,9

LINKS

Alois P. Heinz, Rows n = 0..23, flattened

FORMULA

Sum_{k=1..n} T(n,k) = A296050(n).

EXAMPLE

T(4,0) = 1: 3412.

T(4,1) = 2: 3421, 4312.

T(4,2) = 3: 2413, 3142, 4321.

T(4,3) = 2: 2341, 4123.

T(4,4) = 1: 2143.

Triangle T(n,k) begins:

      1;

      0,     0;

      0,     0,     1;

      0,     0,     2,     0;

      1,     2,     3,     2,     1;

      4,    12,    14,     8,     6,    0;

     29,    68,    82,    54,    25,    6,   1;

    206,   496,   546,   376,   170,   48,  12,   0;

   1708,  3960,  4349,  2922,  1353,  430,  98,  12,  1;

  15702, 35816, 38632, 26048, 12084, 4052, 982, 160, 20, 0;

  ...

MAPLE

b:= proc(s) option remember; expand((n-> `if`(n=0, 1, add(

      (t-> `if`(t=0, 0, `if`(t=1, x, 1)*b(s minus {j}))

       )(abs(n-j)), j=s)))(nops(s)))

    end:

T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b({$1..n})):

seq(T(n), n=0..12);

MATHEMATICA

b[s_] := b[s] = Expand[Function[n, If[n==0, 1, Sum[Function[t, If[t==0, 0, If[t==1, x, 1]*b[s~Complement~{j}]]][Abs[n-j]], {j, s}]]][Length[s]]];

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

T /@ Range[0, 12] // Flatten (* Jean-Fran├žois Alcover, Feb 09 2021, after Alois P. Heinz *)

CROSSREFS

Column k=0 gives A001883.

Row sums give A000166.

Main diagonal and lower diagonal give A059841, A110660.

Cf. A296050, A320582.

Sequence in context: A005813 A049262 A145201 * A340707 A284265 A119464

Adjacent sequences:  A323668 A323669 A323670 * A323672 A323673 A323674

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Jan 23 2019

STATUS

approved

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Last modified August 11 00:55 EDT 2022. Contains 356046 sequences. (Running on oeis4.)