A129534 Triangle read by rows: T(n,k) = number of permutations p of 1,...,n, with min(|p(i)-p(i-1)|, i=2..n) = k (n>=2, k>=1).

2, 6, 22, 2, 106, 14, 630, 88, 2, 4394, 614, 32, 35078, 4874, 366, 2, 315258, 43638, 3912, 72, 3149494, 435002, 42808, 1494, 2, 34620010, 4775184, 496222, 25224, 160, 415222566, 57214716, 6164470, 393792, 6054, 2, 5395737242, 742861262, 82190752, 6070408, 160784, 352

Offset: 2

Author

Emeric Deutsch, May 05 2007

Keywords

• nonn
• tabf

Comments

Row n has floor(n/2) terms. Row sums are the factorial numbers (A000142). T(n,1) = A129535(n). Sum(T(n,k), k>=2) = A002464(n). If, in the definition, min is replaced by max, then one obtains A064482.

Examples

			T(4,2) = 2 because we have 3142 and 2413.
Triangle starts:
     2;
     6;
    22,   2;
   106,  14;
   630,  88,  2;
  4394, 614, 32;
  ...
		

References

• R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 6.40.

Links

• Alois P. Heinz, Rows n = 2..18, flattened

Crossrefs

Cf. A000142, A129535, A002464, A064482.

Programs

Maple

k:=3: with(combinat): a:=proc(n) local P,ct,i: P:=permute(n): ct:=0: for i from 1 to n! do if min(seq(abs(P[i][j]-P[i][j-1]),j=2..n))=k then ct:=ct+1 else ct:=ct: fi: od: ct: end: seq(a(n),n=2..8); # yields the first 7 entries in any specified column k

Extensions

More terms from R. J. Mathar, Oct 11 2007