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

2,1

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.

References

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

Links

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

Example

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

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

Prog

(C++) #include <iostream> #include <vector> #include <algorithm> using namespace std; inline int k(const vector<int> & s) { const int n = s.size() ; int kmin = n+1 ; for(int i=1; i<n; i++) { const int thisdiff = abs(s[i]-s[i-1]) ; if ( thisdiff < kmin) kmin = thisdiff ; } return kmin ; } int main(int argc, char *argv[]) { for(int n=2 ;; n++) { vector<int> s; for(int i=1; i<=n; i++) s.push_back(i) ; vector<unsigned long long > resul(n); do { resul[k(s)]++ ; } while( next_permutation(s.begin(), s.end()) ) ; for(int i=1; i<=n/2; i++) cout << resul[i] << ", " ; cout << endl ; } return 0 ; } - R. J. Mathar, Oct 11 2007

Crossrefs

Cf. A000142, A129535, A002464, A064482.

Sequence in context: A216120 A216964 A187250 * A216719 A085286 A062085

Adjacent sequences: A129531 A129532 A129533 * A129535 A129536 A129537

Keyword

nonn,tabf

Author

Emeric Deutsch, May 05 2007

Extensions

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

Status

approved