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A259784 Number T(n,k) of permutations p of [n] with no fixed points where the maximal displacement of an element equals k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

%I A259784 #25 May 05 2019 08:58:04
%S A259784 1,0,0,0,1,0,0,0,2,0,0,1,3,5,0,0,0,6,18,20,0,0,1,12,44,111,97,0,0,0,
%T A259784 24,116,396,744,574,0,0,1,44,331,1285,3628,5571,3973,0,0,0,84,932,
%U A259784 4312,15038,34948,46662,31520,0,0,1,159,2532,15437,59963,181193,359724,434127,281825,0
%N A259784 Number T(n,k) of permutations p of [n] with no fixed points where the maximal displacement of an element equals k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
%H A259784 Alois P. Heinz, <a href="/A259784/b259784.txt">Rows n = 0..20, flattened</a>
%F A259784 T(n,k) = A259776(n,k) - A259776(n,k-1) for k>0, T(n,0) = A000007(n).
%e A259784 Triangle T(n,k) begins:
%e A259784   1;
%e A259784   0, 0;
%e A259784   0, 1,  0;
%e A259784   0, 0,  2,   0;
%e A259784   0, 1,  3,   5,    0;
%e A259784   0, 0,  6,  18,   20,    0;
%e A259784   0, 1, 12,  44,  111,   97,    0;
%e A259784   0, 0, 24, 116,  396,  744,  574,    0;
%e A259784   0, 1, 44, 331, 1285, 3628, 5571, 3973, 0;
%p A259784 b:= proc(n, s, k) option remember; `if`(n=0, 1, `if`(n+k in s,
%p A259784       b(n-1, (s minus {n+k}) union `if`(n-k>1, {n-k-1}, {}), k),
%p A259784       add(`if`(j=n, 0, b(n-1, (s minus {j}) union
%p A259784       `if`(n-k>1, {n-k-1}, {}), k)), j=s)))
%p A259784     end:
%p A259784 A:= (n, k)-> `if`(k=0, `if`(n=0, 1, 0), b(n, {$max(1, n-k)..n}, k)):
%p A259784 T:= (n, k)-> A(n, k) -`if`(k=0, 0, A(n, k-1)):
%p A259784 seq(seq(T(n, k), k=0..n), n=0..12);
%t A259784 b[n_, s_, k_] := b[n, s, k] = If[n==0, 1, If[MemberQ[s, n+k], b[n-1, (s ~Complement~ {n+k}) ~Union~ If[n-k>1, {n-k-1}, {}], k], Sum[If[j==n, 0, b[n-1, (s ~Complement~ {j}) ~Union~ If[n-k>1, {n-k-1}, {}], k]], {j, s}]] ];
%t A259784 A[n_, k_] := If[k == 0, If[n == 0, 1, 0], b[n, Range[Max[1, n-k], n], k]];
%t A259784 T[n_, k_] :=  A[n, k] - If[k == 0, 0, A[n, k-1]];
%t A259784 Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, May 05 2019, after _Alois P. Heinz_ *)
%Y A259784 Rows sums give A000166.
%Y A259784 Column k=0 and main diagonal give A000007.
%Y A259784 Columns k=1-10 give: A059841 (for n>0), A321048, A321049, A321050, A321051, A321052, A321053, A321054, A321055, A321056.
%Y A259784 First lower diagonal gives A259834.
%Y A259784 T(2n,n) gives A259785.
%Y A259784 Cf. A259776.
%K A259784 nonn,tabl
%O A259784 0,9
%A A259784 _Alois P. Heinz_, Jul 05 2015