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A320582 Number T(n,k) of permutations p of [n] such that |{ j : |p(j)-j| = 1 }| = k; triangle T(n,k), n >= 0, 0 <= k <= n, read by rows. 5
1, 1, 0, 1, 0, 1, 2, 0, 4, 0, 5, 6, 10, 2, 1, 21, 36, 42, 12, 9, 0, 117, 226, 219, 104, 47, 6, 1, 792, 1568, 1472, 800, 328, 64, 16, 0, 6205, 12360, 11596, 6652, 2658, 688, 148, 12, 1, 55005, 109760, 103600, 60840, 24770, 7120, 1560, 200, 25, 0, 543597, 1085560, 1030649, 614420, 255830, 77732, 17750, 2876, 365, 20, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,7

LINKS

Alois P. Heinz, Rows n = 0..24, flattened

FORMULA

Sum_{k=1..n} k * T(n,k) = A052582(n-1) for n > 0.

Sum_{k=0..n} (k+1) * T(n,k) = A082033(n-1) for n > 0.

EXAMPLE

T(4,0) = 5: 1234, 1432, 3214, 3412, 4231.

T(4,1) = 6: 2431, 3241, 3421, 4132, 4213, 4312.

T(4,2) = 10: 1243, 1324, 1342, 1423, 2134, 2314, 2413, 3124, 3142, 4321.

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

T(4,4) = 1: 2143.

Triangle T(n,k) begins:

      1;

      1,      0;

      1,      0,      1;

      2,      0,      4,     0;

      5,      6,     10,     2,     1;

     21,     36,     42,    12,     9,    0;

    117,    226,    219,   104,    47,    6,    1;

    792,   1568,   1472,   800,   328,   64,   16,   0;

   6205,  12360,  11596,  6652,  2658,  688,  148,  12,  1;

  55005, 109760, 103600, 60840, 24770, 7120, 1560, 200, 25,  0;

  ...

MAPLE

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

     `if`(abs(n-j)=1, x, 1)*b(s minus {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);

CROSSREFS

Column k=0 gives A078480.

Row sums give A000142.

Main diagonal gives A059841.

Cf. A008290, A008291, A052582, A082033, A323671.

Sequence in context: A065806 A331185 A228087 * A119690 A166260 A319813

Adjacent sequences:  A320579 A320580 A320581 * A320583 A320584 A320585

KEYWORD

nonn,tabl

AUTHOR

Alois P. Heinz, Jan 23 2019

STATUS

approved

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Last modified May 30 08:43 EDT 2020. Contains 334712 sequences. (Running on oeis4.)