A299789 - OEIS

%I #80 May 01 2019 09:13:07

%S 0,1,1,1,4,2,15,8,1,76,40,4,455,236,28,1,3186,1648,198,8,25487,13125,

%T 1596,111,1,229384,117794,14534,1152,16,2293839,1175224,146372,12929,

%U 435,1,25232230,12903874,1621282,152430,6952,32,302786759,154615096,19563257,1922364,112416,1707,1

%N Number T(n,k) of permutations p of [n] such that min_{j=1..n} |p(j)-j| = k; triangle T(n,k), n >= 0, 0 <= k <= floor(n/2), read by rows.

%H Alois P. Heinz, <a href="/A299789/b299789.txt">Rows n = 0..21, flattened</a>

%F T(n,k) = A306543(n,k) - A306543(n,k+1) for n > 0.

%F Sum_{k=1..floor(n/2)} k * T(n,k) = A129118(n).

%F Sum_{k=1..floor(n/2)} T(n,k) = A000166(n).

%F Sum_{k=2..floor(n/2)} T(n,k) = A001883(n).

%F Sum_{k=3..floor(n/2)} T(n,k) = A075851(n).

%F Sum_{k=4..floor(n/2)} T(n,k) = A075852(n).

%e T(4,0) = 15: 1234, 1243, 1324, 1342, 1423, 1432, 2134, 2314, 2431, 3124, 3214, 3241, 4132, 4213, 4231.

%e T(4,1) = 8: 2143, 2341, 2413, 3142, 3421, 4123, 4312, 4321.

%e T(4,2) = 1: 3412.

%e T(5,2) = 4: 34512, 34521, 45123, 54123.

%e T(6,3) = 1: 456123.

%e T(7,3) = 8: 4567123, 4567132, 4567213, 4567231, 5671234, 5761234, 6571234, 7561234.

%e T(8,4) = 1: 56781234.

%e T(9,4) = 16: 567891234, 567891243, 567891324, 567891342, 567892134, 567892143, 567892314, 567892341, 678912345, 679812345, 687912345, 697812345, 768912345, 769812345, 867912345, 967812345.

%e Triangle T(n,k) begins:

%e           0;

%e           1;

%e           1,         1;

%e           4,         2;

%e          15,         8,        1;

%e          76,        40,        4;

%e         455,       236,       28,       1;

%e        3186,      1648,      198,       8;

%e       25487,     13125,     1596,     111,      1;

%e      229384,    117794,    14534,    1152,     16;

%e     2293839,   1175224,   146372,   12929,    435,    1;

%e    25232230,  12903874,  1621282,  152430,   6952,   32;

%e   302786759, 154615096, 19563257, 1922364, 112416, 1707, 1;

%e   ...

%p b:= proc(s) option remember; (n-> `if`(n=1, x^(s[1]-1),

%p       add((p-> add(coeff(p, x, i)*x^min(i, abs(n-j)),

%p       i=0..degree(p)))(b(s minus {j})), j=s)))(nops(s))

%p     end:

%p T:= n-> (p-> seq(coeff(p, x, i), i=0..max(0, degree(p))))(b({$1..n})):

%p seq(T(n), n=0..14);

%p # second Maple program:

%p A:= proc(n, k) option remember; `if`(n=0, 0, LinearAlgebra[

%p       Permanent](Matrix(n, (i, j)-> `if`(abs(i-j)>=k, 1, 0))))

%p     end:

%p T:= (n, k)-> A(n, k)-A(n, k+1):

%p seq(seq(T(n, k), k=0..n/2), n=0..14);

%t A[n_, k_] := A[n, k] = If[n==0, 0, Permanent[Table[If[Abs[i-j] >= k, 1, 0], {i, 1, n}, {j, 1, n}]]];

%t T[n_, k_] := A[n, k] - A[n, k+1];

%t Table[T[n, k], {n, 0, 14}, {k, 0, n/2}] // Flatten (* _Jean-François Alcover_, May 01 2019, from 2nd Maple program *)

%Y Columns k=0-1 give: A002467, A296050.

%Y Row sums give A000142 (for n>0).

%Y T(2n,n) gives A057427.

%Y T(2n+1,n) gives A000079.

%Y T(2n+2,n) gives A306545.

%Y Cf. A000166, A001883, A075851, A075852, A129118, A130152, A306543.

%K nonn,tabf

%O 0,5

%A _Alois P. Heinz_, Jan 21 2019