A125182 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, 1, 1, 1, 2, 3, 1, 4, 12, 7, 1, 4, 38, 54, 23, 1, 8, 77, 248, 303, 83, 1, 6, 160, 824, 2008, 1636, 405, 1, 11, 285, 2320, 9449, 15789, 10352, 2113, 1, 10, 476, 5564, 37237, 102726, 133293, 70916, 12657, 1, 14, 799, 13172, 122708, 536900, 1158368, 1177168, 537373, 82297

Offset: 1

Author

Emeric Deutsch, Nov 24 2006

Keywords

• nonn
• tabl

Comments

Row sums are the factorial numbers (A000142). T(n,1)=1 (the identity permutation). T(n,2) = A065608(n) = (sum of divisors of n)-(number of divisors of n). T(n,n) = A099152(n). In the first Maple program define n (<=10) to obtain row n.

T(n,k) is also 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). Example: T(4,2)=4 because we have 1432, 3412, 2143 and 3214. - Emeric Deutsch, Nov 28 2008

Examples

			T(4,2) = 4 because we have 4123, 3412, 2143 and 2341.
Triangle starts:
  1;
  1, 1;
  1, 2,  3;
  1, 4, 12,  7;
  1, 4, 38, 54, 23;
  ...
		

Links

• Alois P. Heinz, Rows n = 1..12, flattened
• M. Alekseyev, E. Deutsch, and J. H. Steelman, Problem 11281, Amer. Math. Monthly, 116, No. 5, 2009, p. 465. - _Emeric Deutsch_, Apr 23 2009

Crossrefs

Cf. A000142, A065608, A099152, A125183, A306455.

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(P[j][i]-i,i=1..n)}) = 1 then c[1]:=c[1]+1 elif nops({seq(P[j][i]-i,i=1..n)}) = 2 then c[2]:=c[2]+1 elif nops({seq(P[j][i]-i,i=1..n)}) = 3 then c[3]:=c[3]+1 elif nops({seq(P[j][i]-i,i=1..n)}) = 4 then c[4]:=c[4]+1 elif nops({seq(P[j][i]-i,i=1..n)}) = 5 then c[5]:=c[5]+1 elif nops({seq(P[j][i]-i,i=1..n)}) = 6 then c[6]:=c[6]+1 elif nops({seq(P[j][i]-i,i=1..n)}) = 7 then c[7]:=c[7]+1 elif nops({seq(P[j][i]-i,i=1..n)}) = 8 then c[8]:=c[8]+1 elif nops({seq(P[j][i]-i,i=1..n)}) = 9 then c[9]:=c[9]+1 elif nops({seq(P[j][i]-i,i=1..n)}) = 10 then c[10]:=c[10]+1 else fi od: seq(c[i],i=1..n);
# second Maple program:
b:= proc(p, s) option remember; `if`(p={}, x^nops(s),
      add(b(p minus {t}, s union {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, May 04 2014; revised, Sep 08 2018

Mathematica

b[i_, p_List, s_List] := b[i, p, s] = If[p == {}, x^Length[s], Sum[b[i+1, p ~Complement~ {t}, s ~Union~ {t+i}], {t, p}]]; T[n_] := Function[{p}, Table[ Coefficient[p, x, i], {i, 1, n}]][b[1, Range[n], {}]]; Table[T[n], {n, 1, 9}] // Flatten (* Jean-François Alcover, Feb 19 2015, after Alois P. Heinz *)

Formula

Sum_{k=1..n} k * T(n,k) = A306455(n). - Alois P. Heinz, Feb 16 2019