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

2,4

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)

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:
     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;
  ...

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

Crossrefs

Row sums are A000682. First column is A000560.

Cf. A259701.

Sequence in context: A128704 A075259 A307877 * A316996 A169589 A003570

Adjacent sequences: A259700 A259701 A259702 * A259704 A259705 A259706

Keyword

nonn,tabl

Author

N. J. A. Sloane, Jul 05 2015

Extensions

Corrected and extended by Roger Ford, Jul 06 2016

Status

approved