A375693 -id:A375693 - cp's OEIS frontend

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

Alois P. Heinz, Aug 24 2024

			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, ...

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

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.

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);

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

A375778 Number of multiset permutations of {{1}^n, {2}^n, ..., {n}^n} with no fixed points.

Original entry on oeis.org

1, 0, 1, 56, 748521, 2671644472544, 4165949769769961828425, 4198063809561374304863859278045664, 3792196761630663647644729491214828673313675120817, 4066336002770373278819438293126282402268149361469069425278228336256

Offset: 0

Alois P. Heinz, Aug 27 2024

nonn

			a(0) = 1: the empty permutation.
a(2) = 1: 2211.
a(3) = 56: 222333111, 223133112, 223133121, 223133211, 223313112, 223313121, 223313211, 223331112, 223331121, 223331211, 232133112, 232133121, 232133211, 232313112, 232313121, 232313211, 232331112, 232331121, 232331211, 233113122, 233113212, 233113221, 233131122, 233131212, 233131221, 233311122, 233311212, 233311221, 322133112, 322133121, 322133211, 322313112, 322313121, 322313211, 322331112, 322331121, 322331211, 323113122, 323113212, 323113221, 323131122, 323131212, 323131221, 323311122, 323311212, 323311221, 332113122, 332113212, 332113221, 332131122, 332131212, 332131221, 332311122, 332311212, 332311221, 333111222.

Alois P. Heinz, Table of n, a(n) for n = 0..27

Main diagonal of A372307.

Cf. A059841, A375693.

a:= n-> (-1)^n*int(exp(-x)*orthopoly[L](n, x)^n, x=0..infinity):
seq(a(n), n=0..10);

a[n_] := (-1)^n*Integrate[Exp[-x]*LaguerreL[n, x]^n, {x, 0, Infinity}];
Table[a[n], {n, 0, 10}] (* Jean-François Alcover, Sep 02 2024, after Alois P. Heinz *)

a(n) mod 2 = 1 - (n mod 2) = A059841(n).

cp's OEIS Frontend

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.

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