|
%I #42 Dec 19 2021 12:21:10
%S 1,0,1,0,1,1,0,2,3,1,0,4,11,6,1,0,8,40,35,10,1,0,16,148,195,85,15,1,0,
%T 32,560,1078,665,175,21,1,0,64,2160,5992,5033,1820,322,28,1,0,128,
%U 8448,33632,37632,17913,4284,546,36,1,0,256,33344,190800,280760,171465,52941,9030,870,45,1
%N Number T(n,k) of permutations of [n] having k cycles of the form (c1, c2, ..., c_m) where c1 = min_{i>=1} c_i and c_j = min_{i>=j} c_i or c_j = max_{i>=j} c_i; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
%C The sequence of column k satisfies a linear recurrence with constant coefficients of order k*(k+1)/2 = A000217(k).
%H Alois P. Heinz, <a href="/A344855/b344855.txt">Rows n = 0..140, flattened</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>
%F Sum_{k=1..n} k * T(n,k) = A345341(n).
%F For fixed k, T(n,k) ~ (2*k)^n / (4^k * k!). - _Vaclav Kotesovec_, Jul 15 2021
%e T(4,1) = 4: (1234), (1243), (1423), (1432).
%e Triangle T(n,k) begins:
%e 1;
%e 0, 1;
%e 0, 1, 1;
%e 0, 2, 3, 1;
%e 0, 4, 11, 6, 1;
%e 0, 8, 40, 35, 10, 1;
%e 0, 16, 148, 195, 85, 15, 1;
%e 0, 32, 560, 1078, 665, 175, 21, 1;
%e 0, 64, 2160, 5992, 5033, 1820, 322, 28, 1;
%e ...
%p b:= proc(n) option remember; `if`(n=0, 1, add(expand(x*
%p b(n-j)*binomial(n-1, j-1)*ceil(2^(j-2))), j=1..n))
%p end:
%p T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n)):
%p seq(T(n), n=0..12);
%t b[n_] := b[n] = If[n == 0, 1, Sum[Expand[x*b[n-j]*
%t Binomial[n-1, j-1]*Ceiling[2^(j-2)]], {j, n}]];
%t T[n_] := CoefficientList[b[n], x];
%t Table[T[n], {n, 0, 12}] // Flatten (* _Jean-François Alcover_, Aug 23 2021, after _Alois P. Heinz_ *)
%Y Columns k=0-10 give: A000007, A166444, A346317, A346318, A346319, A346320, A346321, A346322, A346323, A346324, A346325.
%Y Row sums give A187251.
%Y Main diagonal gives A000012, lower diagonal gives A000217, second lower diagonal gives A000914.
%Y T(n+1,n) gives A000217.
%Y T(n+2,n) gives A000914.
%Y T(2n,n) gives A345342.
%Y Cf. A060281, A132393, A186366, A345341.
%K nonn,tabl
%O 0,8
%A _Alois P. Heinz_, May 30 2021
|
