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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. 6
 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, 216, 294, 358, 411, 455, 411, 358, 294, 216, 120, 720, 1320, 1824, 2250, 2612, 2921, 3186, 2921, 2612, 2250, 1824, 1320, 720, 5040, 9360, 13080, 16296, 19086, 21514, 23633, 25487, 23633, 21514, 19086, 16296, 13080, 9360, 5040 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 LINKS Alois P. Heinz, Rows n = 1..142, flattened Wikipedia, Permutation FORMULA T(n,k) = T(n,-k). T(n,k) = - Sum_{j=1..n} (-1)^j * binomial(n-|k|,j) * (n-j)!. T(n,k) = |k|! * (n-|k|)! [x^(n-|k|)] (1-exp(-x))/(1-x)^(|k|+1). Sum_{k=1-n..n-1} T(n,k) = A306455(n). T(n,k) = |k|! * A306234(n,k). EXAMPLE 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]. Triangle T(n,k) begins:   :                             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, 216, 294, 358, 411, 455, 411, 358, 294, 216, 120  ; MAPLE b:= proc(s, d) option remember; (n-> `if`(n=0, add(x^j, j=d),       add(b(s minus {i}, d union {n-i}), i=s)))(nops(s))     end: T:= n-> (p-> seq(coeff(p, x, i), i=1-n..n-1))(b({\$1..n}, {})): seq(T(n), n=1..8); # second Maple program: T:= (n, k)-> -add((-1)^j*binomial(n-abs(k), j)*(n-j)!, j=1..n): seq(seq(T(n, k), k=1-n..n-1), n=1..9); CROSSREFS Columns k=0-1 give: A002467, A180191. Row sums give A306455. T(n+1,n) gives A000142. T(n+2,n) gives A007680. Cf. A000142, A061018 (left half of this triangle), A306234, A306506, A324225. Sequence in context: A251102 A217287 A111880 * A101497 A274007 A065870 Adjacent sequences:  A306458 A306459 A306460 * A306462 A306463 A306464 KEYWORD nonn,tabf AUTHOR Alois P. Heinz, Feb 17 2019 STATUS approved

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Last modified January 29 01:54 EST 2020. Contains 331328 sequences. (Running on oeis4.)