A259703 Triangle read by rows: T(n,k) = number of permutations without overlaps in which the first increasing run has length k.

1, 1, 1, 2, 1, 1, 5, 2, 2, 1, 12, 5, 4, 2, 1, 33, 13, 12, 4, 3, 1, 87, 35, 30, 12, 6, 3, 1, 252, 98, 90, 32, 21, 6, 4, 1, 703, 278, 243, 94, 54, 21, 8, 4, 1, 2105, 812, 745, 270, 175, 57, 32, 8, 5, 1, 6099, 2385, 2108, 808, 485, 181, 84, 32, 10, 5, 1

Offset: 2

Author

N. J. A. Sloane, Jul 05 2015

Keywords

• nonn
• tabl

Comments

The 12th row of the triangle (as given in the reference) is definitely wrong, since the first column of this triangle is known (it is A000560). The row sums are also known - see A000682.

From Roger Ford, Jul 06 2016: (Start)

To determine the first increasing run of the permutation 176852943 start on the left and move to the right counting the consecutive integers.

(1)7685(2)94(3). This permutation a has a first run of (3-1)=2. The permutation 123465 has a first run of (5-1)=4. (1)(2)(3)(4)6(5). (End)

Examples

			Triangle begins:
     1;
     1,    1;
     2,    1,    1;
     5,    2,    2,   1;
    12,    5,    4,   2,   1;
    33,   13,   12,   4,   3,   1;
    87,   35,   30,  12,   6,   3,  1;
   252,   98,   90,  32,  21,   6,  4,  1;
   703,  278,  243,  94,  54,  21,  8,  4,  1;
  2105,  812,  745, 270, 175,  57, 32,  8,  5, 1;
  6099, 2385, 2108, 808, 485, 181, 84, 32, 10, 5, 1;
  ...
		

References

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

Links

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

Crossrefs

Row sums are A000682. First column is A000560.

Cf. A259701.

Programs

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(Overlapfree(Chords(v)), r[FirstRunLen(v)]++))); r}
for(n=2, 8, print(row(n))) \\ Andrew Howroyd, Dec 07 2018

Extensions

Corrected and extended by Roger Ford, Jul 06 2016