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A299789 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.

%I A299789 #80 May 01 2019 09:13:07
%S A299789 0,1,1,1,4,2,15,8,1,76,40,4,455,236,28,1,3186,1648,198,8,25487,13125,
%T A299789 1596,111,1,229384,117794,14534,1152,16,2293839,1175224,146372,12929,
%U A299789 435,1,25232230,12903874,1621282,152430,6952,32,302786759,154615096,19563257,1922364,112416,1707,1
%N A299789 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 A299789 Alois P. Heinz, <a href="/A299789/b299789.txt">Rows n = 0..21, flattened</a>
%F A299789 T(n,k) = A306543(n,k) - A306543(n,k+1) for n > 0.
%F A299789 Sum_{k=1..floor(n/2)} k * T(n,k) = A129118(n).
%F A299789 Sum_{k=1..floor(n/2)} T(n,k) = A000166(n).
%F A299789 Sum_{k=2..floor(n/2)} T(n,k) = A001883(n).
%F A299789 Sum_{k=3..floor(n/2)} T(n,k) = A075851(n).
%F A299789 Sum_{k=4..floor(n/2)} T(n,k) = A075852(n).
%e A299789 T(4,0) = 15: 1234, 1243, 1324, 1342, 1423, 1432, 2134, 2314, 2431, 3124, 3214, 3241, 4132, 4213, 4231.
%e A299789 T(4,1) = 8: 2143, 2341, 2413, 3142, 3421, 4123, 4312, 4321.
%e A299789 T(4,2) = 1: 3412.
%e A299789 T(5,2) = 4: 34512, 34521, 45123, 54123.
%e A299789 T(6,3) = 1: 456123.
%e A299789 T(7,3) = 8: 4567123, 4567132, 4567213, 4567231, 5671234, 5761234, 6571234, 7561234.
%e A299789 T(8,4) = 1: 56781234.
%e A299789 T(9,4) = 16: 567891234, 567891243, 567891324, 567891342, 567892134, 567892143, 567892314, 567892341, 678912345, 679812345, 687912345, 697812345, 768912345, 769812345, 867912345, 967812345.
%e A299789 Triangle T(n,k) begins:
%e A299789           0;
%e A299789           1;
%e A299789           1,         1;
%e A299789           4,         2;
%e A299789          15,         8,        1;
%e A299789          76,        40,        4;
%e A299789         455,       236,       28,       1;
%e A299789        3186,      1648,      198,       8;
%e A299789       25487,     13125,     1596,     111,      1;
%e A299789      229384,    117794,    14534,    1152,     16;
%e A299789     2293839,   1175224,   146372,   12929,    435,    1;
%e A299789    25232230,  12903874,  1621282,  152430,   6952,   32;
%e A299789   302786759, 154615096, 19563257, 1922364, 112416, 1707, 1;
%e A299789   ...
%p A299789 b:= proc(s) option remember; (n-> `if`(n=1, x^(s[1]-1),
%p A299789       add((p-> add(coeff(p, x, i)*x^min(i, abs(n-j)),
%p A299789       i=0..degree(p)))(b(s minus {j})), j=s)))(nops(s))
%p A299789     end:
%p A299789 T:= n-> (p-> seq(coeff(p, x, i), i=0..max(0, degree(p))))(b({$1..n})):
%p A299789 seq(T(n), n=0..14);
%p A299789 # second Maple program:
%p A299789 A:= proc(n, k) option remember; `if`(n=0, 0, LinearAlgebra[
%p A299789       Permanent](Matrix(n, (i, j)-> `if`(abs(i-j)>=k, 1, 0))))
%p A299789     end:
%p A299789 T:= (n, k)-> A(n, k)-A(n, k+1):
%p A299789 seq(seq(T(n, k), k=0..n/2), n=0..14);
%t A299789 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 A299789 T[n_, k_] := A[n, k] - A[n, k+1];
%t A299789 Table[T[n, k], {n, 0, 14}, {k, 0, n/2}] // Flatten (* _Jean-François Alcover_, May 01 2019, from 2nd Maple program *)
%Y A299789 Columns k=0-1 give: A002467, A296050.
%Y A299789 Row sums give A000142 (for n>0).
%Y A299789 T(2n,n) gives A057427.
%Y A299789 T(2n+1,n) gives A000079.
%Y A299789 T(2n+2,n) gives A306545.
%Y A299789 Cf. A000166, A001883, A075851, A075852, A129118, A130152, A306543.
%K A299789 nonn,tabf
%O A299789 0,5
%A A299789 _Alois P. Heinz_, Jan 21 2019