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A344901 Triangle read by rows: T(n,k) is the number of permutations of length n that have k same elements at the same positions with its inverse permutation for 0 <= k <= n.

%I A344901 #37 Apr 24 2025 06:15:17
%S A344901 1,0,1,0,0,2,2,0,0,4,6,8,0,0,10,24,30,40,0,0,26,160,144,180,160,0,0,
%T A344901 76,1140,1120,1008,840,700,0,0,232,8988,9120,8960,5376,4200,2912,0,0,
%U A344901 764,80864,80892,82080,53760,30240,19656,12768,0,0,2620,809856,808640,808920,547200,336000,157248,95760,55680,0,0,9496
%N A344901 Triangle read by rows: T(n,k) is the number of permutations of length n that have k same elements at the same positions with its inverse permutation for 0 <= k <= n.
%H A344901 Alois P. Heinz, <a href="/A344901/b344901.txt">Rows n = 0..140, flattened</a>
%F A344901 T(n,k) = binomial(n,k)*A000085(k)*A038205(n-k).
%F A344901 From _Alois P. Heinz_, Oct 28 2024: (Start)
%F A344901 Sum_{k=0..n} k * T(n,k) = A052849(n) = A098558(n) for n>=2.
%F A344901 Sum_{k=0..n} (n-k) * T(n,k) = A052571(n).
%F A344901 Sum_{k=0..n} (-1)^k * T(n,k) = A000023(n).
%F A344901 T(n,0) + T(n,1) = A137482(n). (End)
%e A344901 Triangle T(n,k) begins:
%e A344901      1;
%e A344901      0,    1;
%e A344901      0,    0,    2;
%e A344901      2,    0,    0,    4;
%e A344901      6,    8,    0,    0,   10;
%e A344901     24,   30,   40,    0,    0,   26;
%e A344901    160,  144,  180,  160,    0,    0, 76;
%e A344901   1140, 1120, 1008,  840,  700,    0,  0, 232;
%e A344901   8988, 9120, 8960, 5376, 4200, 2912,  0,   0, 764;
%e A344901   ...
%p A344901 b:= proc(n, t) option remember; `if`(n=0, 1, add(b(n-j, t)*
%p A344901       binomial(n-1, j-1)*(j-1)!, j=`if`(t=1, 1..min(2, n), 3..n)))
%p A344901     end:
%p A344901 T:= (n, k)-> binomial(n, k)*b(k, 1)*b(n-k, 0):
%p A344901 seq(seq(T(n, k), k=0..n), n=0..10);  # _Alois P. Heinz_, Oct 28 2024
%t A344901 b[n_, t_] := b[n, t] = If[n == 0, 1, Sum[b[n-j, t]* Binomial[n-1, j-1]*(j-1)!, {j, If[t == 1, Range @ Min[2, n], Range[3, n]]}]];
%t A344901 T[n_, k_] := Binomial[n, k]*b[k, 1]*b[n-k, 0];
%t A344901 Table[Table[T[n, k], {k, 0, n}], {n, 0, 10}] // Flatten (* _Jean-François Alcover_, Apr 24 2025, after _Alois P. Heinz_ *)
%Y A344901 Columns k=0-1 give: A038205, A221145.
%Y A344901 Row sums give A000142.
%Y A344901 Main diagonal gives A000085.
%Y A344901 Cf. A000023, A007318, A052571, A052849, A098558, A137482.
%K A344901 nonn,tabl
%O A344901 0,6
%A A344901 _Mikhail Kurkov_, Jun 01 2021