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.

%I A375694 #28 Aug 27 2024 12:05:41
%S A375694 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,
%T A375694 0,251,34442,362997,101644,265,0,1,0,923,756002,62924817,166336604,
%U A375694 6840085,1854,0,1,0,3431,17150366,11729719509,305225265804,136221590695,630985830,14833,0
%N 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.
%H A375694 Alois P. Heinz, <a href="/A375694/b375694.txt">Antidiagonals n = 0..53, flattened</a>
%F A375694 A(n,k) = Sum_{j=0..n} (-1)^(n-j)*binomial(n,j)*(k*j)!/k!^j.
%e A375694 A(2,2) = 5: 1212, 1221, 2112, 2121, 2211.
%e A375694 A(2,3) = 19: 112122, 112212, 112221, 121122, 121212, 121221, 122112, 122121, 122211, 211122, 211212, 211221, 212112, 212121, 212211, 221112, 221121, 221211, 222111.
%e A375694 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.
%e A375694 A(4,1) = 9: 2143, 2341, 2413, 3142, 3412, 3421, 4123, 4312, 4321.
%e A375694 Square array A(n,k) begins:
%e A375694   1,  1,      1,         1,            1,               1, ...
%e A375694   0,  0,      0,         0,            0,               0, ...
%e A375694   0,  1,      5,        19,           69,             251, ...
%e A375694   0,  2,     74,      1622,        34442,          756002, ...
%e A375694   0,  9,   2193,    362997,     62924817,     11729719509, ...
%e A375694   0, 44, 101644, 166336604, 305225265804, 623302086965044, ...
%p A375694 A:= (n, k)-> add((-1)^(n-j)*binomial(n, j)*(k*j)!/k!^j, j=0..n):
%p A375694 seq(seq(A(n, d-n), n=0..d), d=0..10);
%Y A375694 Columns k=0-2 give: A000007, A000166, A374980.
%Y A375694 Rows n=0-2 give: A000012, A000004, A030662.
%Y A375694 Main diagonal gives A375693.
%Y A375694 Cf. A060538, A089759, A187783, A372307.
%K A375694 nonn,tabl
%O A375694 0,13
%A A375694 _Alois P. Heinz_, Aug 24 2024