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 A238889 Number T(n,k) of self-inverse permutations p on [n] where the maximal displacement of an element equals k: k = max_{i=1..n} |p(i)-i|; triangle T(n,k), n>=0, 0<=k<=n, read by rows. 11
 1, 1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 4, 3, 2, 0, 1, 7, 7, 7, 4, 0, 1, 12, 16, 19, 18, 10, 0, 1, 20, 35, 47, 55, 48, 26, 0, 1, 33, 74, 117, 151, 170, 142, 76, 0, 1, 54, 153, 284, 399, 515, 544, 438, 232, 0, 1, 88, 312, 675, 1061, 1471, 1826, 1846, 1452, 764, 0, 1, 143, 629, 1575, 2792, 4119, 5651, 6664, 6494, 5008, 2620, 0 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,8 COMMENTS Main diagonal and lower diagonal give: A000007, A000085(n-1). Columns k=0-10 give: A000012, A000071(n+1), A238913, A238914, A238915, A238916, A238917, A238918, A238919, A238920, A238921. Row sums give A000085. LINKS Joerg Arndt and Alois P. Heinz, Rows n=0..28, flattened FORMULA T(n,k) = A238888(n,k) - A238888(n,k-1) for k>0, T(n,0) = 1. EXAMPLE T(4,0) = 1: 1234. T(4,1) = 4: 1243, 1324, 2134, 2143. T(4,2) = 3: 1432, 3214, 3412. T(4,3) = 2: 4231, 4321. Triangle T(n,k) begins: 00: 1; 01: 1,   0; 02: 1,   1,   0; 03: 1,   2,   1,   0; 04: 1,   4,   3,   2,    0; 05: 1,   7,   7,   7,    4,    0; 06: 1,  12,  16,  19,   18,   10,    0; 07: 1,  20,  35,  47,   55,   48,   26,    0; 08: 1,  33,  74, 117,  151,  170,  142,   76,    0; 09: 1,  54, 153, 284,  399,  515,  544,  438,  232,   0; 10: 1,  88, 312, 675, 1061, 1471, 1826, 1846, 1452, 764,  0; ... The 26 involutions of 5 elements together with their maximal displacements are: 01:  [ 1 2 3 4 5 ]   0 02:  [ 1 2 3 5 4 ]   1 03:  [ 1 2 4 3 5 ]   1 04:  [ 1 2 5 4 3 ]   2 05:  [ 1 3 2 4 5 ]   1 06:  [ 1 3 2 5 4 ]   1 07:  [ 1 4 3 2 5 ]   2 08:  [ 1 4 5 2 3 ]   2 09:  [ 1 5 3 4 2 ]   3 10:  [ 1 5 4 3 2 ]   3 11:  [ 2 1 3 4 5 ]   1 12:  [ 2 1 3 5 4 ]   1 13:  [ 2 1 4 3 5 ]   1 14:  [ 2 1 5 4 3 ]   2 15:  [ 3 2 1 4 5 ]   2 16:  [ 3 2 1 5 4 ]   2 17:  [ 3 4 1 2 5 ]   2 18:  [ 3 5 1 4 2 ]   3 19:  [ 4 2 3 1 5 ]   3 20:  [ 4 2 5 1 3 ]   3 21:  [ 4 3 2 1 5 ]   3 22:  [ 4 5 3 1 2 ]   3 23:  [ 5 2 3 4 1 ]   4 24:  [ 5 2 4 3 1 ]   4 25:  [ 5 3 2 4 1 ]   4 26:  [ 5 4 3 2 1 ]   4 There is one involution with no displacements, 7 with one displacement, etc. giving row 4: [1, 7, 7, 7, 4, 0]. MAPLE b:= proc(n, k, s) option remember; `if`(n=0, 1, `if`(n in s,       b(n-1, k, s minus {n}), b(n-1, k, s) +add(`if`(i in s, 0,       b(n-1, k, s union {i})), i=max(1, n-k)..n-1)))     end: A:= (n, k)-> `if`(k<0, 0, b(n, k, {})): T:= (n, k)-> A(n, k) -A(n, k-1): seq(seq(T(n, k), k=0..n), n=0..14); MATHEMATICA b[n_, k_, s_List] := b[n, k, s] = If[n == 0, 1, If[MemberQ[s, n], b[n-1, k, DeleteCases[s, n]], b[n-1, k, s] + Sum[If[MemberQ[s, i], 0, b[n-1, k, s ~Union~ {i}]], {i, Max[1, n-k], n-1}]]]; A[n_, k_] := If[k<0, 0, b[n, k, {}]]; T[n_, k_] := A[n, k] - A[n, k-1]; Table[Table[T[n, k], {k, 0, n}], {n, 0, 14}] // Flatten (* Jean-François Alcover, Jan 08 2015, translated from Maple *) CROSSREFS Sequence in context: A088455 A004248 A034373 * A296207 A253628 A102728 Adjacent sequences:  A238886 A238887 A238888 * A238890 A238891 A238892 KEYWORD nonn,tabl AUTHOR Joerg Arndt and Alois P. Heinz, Mar 06 2014 STATUS approved

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Last modified August 11 00:32 EDT 2020. Contains 336403 sequences. (Running on oeis4.)