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
Examples
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, ...
Links
- Alois P. Heinz, Antidiagonals n = 0..53, flattened
Crossrefs
Programs
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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);
Formula
A(n,k) = Sum_{j=0..n} (-1)^(n-j)*binomial(n,j)*(k*j)!/k!^j.