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A306461 Number T(n,k) of occurrences of k in a (signed) displacement set of a permutation of [n]; triangle T(n,k), n>=1, 1-n<=k<=n-1, read by rows.

%I A306461 #48 Feb 20 2021 03:36:12
%S A306461 1,1,1,1,2,3,4,3,2,6,10,13,15,13,10,6,24,42,56,67,76,67,56,42,24,120,
%T A306461 216,294,358,411,455,411,358,294,216,120,720,1320,1824,2250,2612,2921,
%U A306461 3186,2921,2612,2250,1824,1320,720,5040,9360,13080,16296,19086,21514,23633,25487,23633,21514,19086,16296,13080,9360,5040
%N A306461 Number T(n,k) of occurrences of k in a (signed) displacement set of a permutation of [n]; triangle T(n,k), n>=1, 1-n<=k<=n-1, read by rows.
%H A306461 Alois P. Heinz, <a href="/A306461/b306461.txt">Rows n = 1..142, flattened</a>
%H A306461 Wikipedia, <a href="https://en.wikipedia.org/wiki/Permutation">Permutation</a>
%F A306461 T(n,k) = T(n,-k).
%F A306461 T(n,k) = - Sum_{j=1..n} (-1)^j * binomial(n-|k|,j) * (n-j)!.
%F A306461 T(n,k) = |k|! * (n-|k|)! [x^(n-|k|)] (1-exp(-x))/(1-x)^(|k|+1).
%F A306461 Sum_{k=1-n..n-1} T(n,k) = A306455(n).
%F A306461 T(n,k) = |k|! * A306234(n,k).
%e A306461 The 6 permutations p of [3]: 123, 132, 213, 231, 312, 321 have (signed) displacement sets {p(i)-i, i=1..3}: {0}, {-1,0,1}, {-1,0,1}, {-2,1}, {-1,2}, {-2,0,2}, respectively. Numbers -2 and 2 occur twice, -1 and 1 occur thrice, and 0 occurs four times. So row n=3 is [2, 3, 4, 3, 2].
%e A306461 Triangle T(n,k) begins:
%e A306461   :                             1                           ;
%e A306461   :                        1,   1,   1                      ;
%e A306461   :                   2,   3,   4,   3,   2                 ;
%e A306461   :              6,  10,  13,  15,  13,  10,   6            ;
%e A306461   :        24,  42,  56,  67,  76,  67,  56,  42,  24       ;
%e A306461   :  120, 216, 294, 358, 411, 455, 411, 358, 294, 216, 120  ;
%p A306461 b:= proc(s, d) option remember; (n-> `if`(n=0, add(x^j, j=d),
%p A306461       add(b(s minus {i}, d union {n-i}), i=s)))(nops(s))
%p A306461     end:
%p A306461 T:= n-> (p-> seq(coeff(p, x, i), i=1-n..n-1))(b({$1..n}, {})):
%p A306461 seq(T(n), n=1..8);
%p A306461 # second Maple program:
%p A306461 T:= (n, k)-> -add((-1)^j*binomial(n-abs(k), j)*(n-j)!, j=1..n):
%p A306461 seq(seq(T(n, k), k=1-n..n-1), n=1..9);
%t A306461 T[n_, k_] := -Sum[(-1)^j Binomial[n-Abs[k], j] (n-j)!, {j, 1, n}];
%t A306461 Table[Table[T[n, k], {k, 1-n, n-1}], {n, 1, 9}] // Flatten (* _Jean-François Alcover_, Feb 20 2021, after _Alois P. Heinz_ *)
%Y A306461 Columns k=0-1 give: A002467, A180191.
%Y A306461 Row sums give A306455.
%Y A306461 T(n+1,n) gives A000142.
%Y A306461 T(n+2,n) gives A007680.
%Y A306461 Cf. A000142, A061018 (left half of this triangle), A306234, A306506, A324225.
%K A306461 nonn,tabf
%O A306461 1,5
%A A306461 _Alois P. Heinz_, Feb 17 2019