A125183 Triangle read by rows: T(n,k) is the number of permutations p of {1,2,...,n} such that the set {|p(i)-i|, i=1,2,...,n} has exactly k elements (1<=k<=n).

1, 2, 0, 1, 5, 0, 3, 11, 6, 4, 1, 28, 55, 32, 4, 3, 69, 210, 330, 108, 0, 1, 102, 846, 2177, 1590, 324, 0, 4, 279, 2694, 11221, 17578, 7624, 888, 32, 1, 328, 7791, 54777, 135993, 123474, 37524, 2896, 96, 3, 961, 24032, 227906, 914364, 1427342, 839904, 182824, 11464, 0

Offset: 1

Author

Emeric Deutsch, Nov 24 2006

Keywords

• nonn
• tabl

Comments

Row sums are the factorial numbers (A000142). T(n,n) = A075866(n). In the Maple program define n (<=10) to obtain row n.

Examples

			T(4,3) = 6 because we have 1423, 1342, 3124, 4312, 2314 and 3421.
Triangle starts:
  1;
  2,  0;
  1,  5,   0;
  3, 11,   6,   4;
  1, 28,  55,  32,   4;
  3, 69, 210, 330, 108,  0;
  ...
		

Links

• Alois P. Heinz, Rows n = 1..14, flattened

Crossrefs

Cf. A000142, A075866, A125182.

Programs

Maple

n:=7: with(combinat): P:=permute(n): for j from 1 to n! do c[j]:=0 od: for j from 1 to n! do if nops({seq(abs(P[j][i]-i),i=1..n)}) = 1 then c[1]:=c[1]+1 elif nops({seq(abs(P[j][i]-i),i=1..n)}) = 2 then c[2]:=c[2]+1 elif nops({seq(abs(P[j][i]-i),i=1..n)}) = 3 then c[3]:=c[3]+1 elif nops({seq(abs(P[j][i]-i),i=1..n)}) = 4 then c[4]:=c[4]+1 elif nops({seq(abs(P[j][i]-i),i=1..n)}) = 5 then c[5]:=c[5]+1 elif nops({seq(abs(P[j][i]-i),i=1..n)}) = 6 then c[6]:=c[6]+1 elif nops({seq(abs(P[j][i]-i),i=1..n)}) = 7 then c[7]:=c[7]+1 elif nops({seq(abs(P[j][i]-i),i=1..n)}) = 8 then c[8]:=c[8]+1 elif nops({seq(abs(P[j][i]-i),i=1..n)}) = 9 then c[9]:=c[9]+1 elif nops({seq(abs(P[j][i]-i),i=1..n)}) = 10 then c[10]:=c[10]+1 else fi od: seq(c[i],i=1..n); # yields row n for the specified n (n<=10)
# second Maple program:
b:= proc(p, s) option remember; `if`(p={}, x^nops(s),
      add(b(p minus {t}, s union {abs(t-nops(p))}), t=p))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..n))(b({$1..n}, {})):
seq(T(n), n=1..9);  # Alois P. Heinz, Feb 21 2019

Mathematica

b[p_, s_] := b[p, s] = If[p == {}, x^Length[s], Sum[b[p ~Complement~ {t}, s ~Union~ {Abs[t - Length[p]]}], {t, p}]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, n}]][b[Range[n], {}]];
Table[T[n], {n, 1, 10}] // Flatten (* Jean-François Alcover, Dec 07 2019, after Alois P. Heinz *)

Extensions

More terms from Alois P. Heinz, Feb 27 2012