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 A062866 Triangle of number of permutations by barycenter. 10
 1, 1, 2, 1, 4, 1, 1, 4, 14, 4, 1, 1, 5, 31, 46, 31, 5, 1, 1, 6, 66, 146, 282, 146, 66, 6, 1, 1, 7, 134, 392, 1289, 1394, 1289, 392, 134, 7, 1, 1, 8, 267, 960, 4859, 7736, 12658, 7736, 4859, 960, 267, 8, 1, 1, 9, 529, 2235, 16615, 34659, 85831, 83122, 85831, 34659, 16615, 2235, 529, 9, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS The barycenter or signcenter of a permutation is the sum of the signs of the difference between initial and final positions of the objects. LINKS Alois P. Heinz, Rows n = 0..20, flattened FORMULA From Alois P. Heinz, Jul 31 2018: (Start) T(n,k) = T(n,-k). Sum_{k>=0} T(n,k) = A179566(n). (End) EXAMPLE (1,3,2,5,7,6,4) has difference (0,1,-1,1,2,0,-3) and signs (0,1,-1,1,1,0,-1) with total 1. This is one of 1289 such permutations of degree 7. Triangle begins: :                                  1                              ; :                                  1                              ; :                                  2                              ; :                           1,     4,    1                        ; :                     1,    4,    14,    4,    1                  ; :               1,    5,   31,    46,   31,    5,   1             ; :          1,   6,   66,  146,   282,  146,   66,   6,   1        ; :     1,   7, 134,  392, 1289,  1394, 1289,  392, 134,   7, 1     ; :  1, 8, 267, 960, 4859, 7736, 12658, 7736, 4859, 960, 267, 8, 1  ; MAPLE b:= proc(s, t) option remember; (n-> `if`(n=0, x^t,       add(b(s minus {j}, t+signum(n-j)), j=s)))(nops(s))     end: T:= n-> (p-> seq(coeff(p, x, i), i=ldegree(p)..degree(p)))(b({\$1..n}, 0)): seq(T(n), n=0..11);  # Alois P. Heinz, Jul 31 2018 MATHEMATICA row[n_] := Sort[Tally[Total[Sign[# - Range[n]]]& /@ Permutations[Range[n]] ]][[All, 2]]; Array[row, 9] // Flatten (* Jean-François Alcover, Oct 07 2016 *) CROSSREFS Columns k=0-4 give: A062868, A179562, A169934, A179564, A179565. Row sums give A000142. Cf. A062867, A179566. Sequence in context: A255878 A162944 A140582 * A131035 A118745 A235671 Adjacent sequences:  A062863 A062864 A062865 * A062867 A062868 A062869 KEYWORD nice,nonn,tabf AUTHOR Olivier Gérard, Jun 26 2001 STATUS approved

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Last modified September 21 10:03 EDT 2020. Contains 337268 sequences. (Running on oeis4.)