A259701 Triangle read by rows: T(n,k) = number of permutations without overlaps in which the first increasing run has length k and the second element is not 2.

0, 1, 0, 2, 0, 0, 5, 0, 1, 0, 12, 0, 2, 0, 0, 33, 1, 7, 0, 1, 0, 87, 2, 17, 0, 2, 0, 0, 252, 11, 55, 2, 9, 0, 1, 0, 703, 26, 145, 4, 22, 0, 2, 0, 0, 2105, 109, 467, 27, 81, 3, 11, 0, 1, 0, 6099, 280, 1296, 63, 215, 6, 27, 0, 2, 0, 0

Offset

2,4

Comments

The 12th row of the triangle given in the Sade reference is incorrect, since the first column of this triangle is known (it is A000560).

References

A. Sade, Sur les Chevauchements des Permutations, published by the author, Marseille, 1949

Links

Table of n, a(n) for n=2..67.

Albert Sade, Sur les Chevauchements des Permutations, published by the author, Marseille, 1949. [Annotated scanned copy]

Example

Triangle begins:
     0;
     1,   0;
     2,   0,   0;
     5,   0,   1,   0;
    12,   0,   2,   0,  0;
    33,   1,   7,   0,  1, 0;
    87,   2,  17,   0,  2, 0,  0;
   252,  11,  55,   2,  9, 0,  1, 0;
   703,  26, 145,   4, 22, 0,  2, 0, 0;
  2105, 109, 467,  27, 81, 3, 11, 0, 1, 0;
  ...

Prog

(PARI)
Overlapfree(v)={for(i=1, #v, for(j=i+1, v[i]-1, if(v[j]>v[i], return(0)))); 1}
Chords(u)={my(n=2*#u, v=vector(n), s=u[#u]); if(s%2==0, s=n+1-s); for(i=1, #u, my(t=n+1-s); s=u[i]; if(s%2==0, s=n+1-s); v[s]=t; v[t]=s); v}
FirstRunLen(v)={my(e=1); for(i=1, #v, if(v[i]==e, e++)); e-2}
row(n)={my(r=vector(n-1)); if(n>=2, forperm(n, v, if(v[1]<>1, break); if(v[2]<>2 && Overlapfree(Chords(v)), r[FirstRunLen(v)]++))); r}
for(n=2, 8, print(row(n))) \\ Andrew Howroyd, Dec 07 2018

Crossrefs

Row sums excluding the first column give A259702.

First column is A000560.

Cf. A259703.

Sequence in context: A289088 A057611 A329959 * A147843 A094597 A202992

Adjacent sequences: A259698 A259699 A259700 * A259702 A259703 A259704

Keyword

nonn,tabl,more

Author

N. J. A. Sloane, Jul 05 2015

Extensions

a(49) corrected and a(57)-a(67) from Andrew Howroyd, Dec 07 2018

Status

approved