A375694 Number A(n,k) of multiset permutations of {{1}^k, {2}^k, ..., {n}^k} with no fixed k-tuple {j}^k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

1, 1, 0, 1, 0, 0, 1, 0, 1, 0, 1, 0, 5, 2, 0, 1, 0, 19, 74, 9, 0, 1, 0, 69, 1622, 2193, 44, 0, 1, 0, 251, 34442, 362997, 101644, 265, 0, 1, 0, 923, 756002, 62924817, 166336604, 6840085, 1854, 0, 1, 0, 3431, 17150366, 11729719509, 305225265804, 136221590695, 630985830, 14833, 0

Offset

0,13

Links

Alois P. Heinz, Antidiagonals n = 0..53, flattened

Formula

A(n,k) = Sum_{j=0..n} (-1)^(n-j)*binomial(n,j)*(k*j)!/k!^j.

Example

A(2,2) = 5: 1212, 1221, 2112, 2121, 2211.
A(2,3) = 19: 112122, 112212, 112221, 121122, 121212, 121221, 122112, 122121, 122211, 211122, 211212, 211221, 212112, 212121, 212211, 221112, 221121, 221211, 222111.
A(3,2) = 74: 121323, 121332, 122313, 122331, 123123, 123132, 123213, 123231, 123312, 123321, 131223, 131232, 131322, 132123, 132132, 132312, 132321, 133122, 133212, 133221, 211323, 211332, 212313, 212331, 213123, 213132, 213213, 213231, 213312, 213321, 221313, 221331, 223113, 223131, 223311, 231123, 231132, 231213, 231231, 231312, 231321, 232113, 232131, 232311, 233112, 233121, 233211, 311223, 311232, 311322, 312123, 312132, 312312, 312321, 313122, 313212, 313221, 321123, 321132, 321213, 321231, 321312, 321321, 322113, 322131, 322311, 323112, 323121, 323211, 331122, 331212, 331221, 332112, 332121.
A(4,1) = 9: 2143, 2341, 2413, 3142, 3412, 3421, 4123, 4312, 4321.
Square array A(n,k) begins:
  1,  1,      1,         1,            1,               1, ...
  0,  0,      0,         0,            0,               0, ...
  0,  1,      5,        19,           69,             251, ...
  0,  2,     74,      1622,        34442,          756002, ...
  0,  9,   2193,    362997,     62924817,     11729719509, ...
  0, 44, 101644, 166336604, 305225265804, 623302086965044, ...

Maple

A:= (n, k)-> add((-1)^(n-j)*binomial(n, j)*(k*j)!/k!^j, j=0..n):
seq(seq(A(n, d-n), n=0..d), d=0..10);

Crossrefs

Columns k=0-2 give: A000007, A000166, A374980.

Rows n=0-2 give: A000012, A000004, A030662.

Main diagonal gives A375693.

Cf. A060538, A089759, A187783, A372307.

Sequence in context: A249421 A359226 A369286 * A244813 A078110 A334708

Adjacent sequences: A375691 A375692 A375693 * A375695 A375696 A375697

Keyword

nonn,tabl

Author

Alois P. Heinz, Aug 24 2024

Status

approved