A259701 -id:A259701 - cp's OEIS frontend

A259702 Row sums of A259701 except first column.

0, 0, 0, 1, 2, 9, 21, 78, 199, 699, 1889, 6491, 18261, 62145, 180091, 610220, 1809045, 6118849, 18469079, 62440111, 191235803, 646681908, 2004592956, 6782895492, 21239394216, 71925883149, 227169634741, 769998727785, 2450045838331, 8312417389237, 26620229804149

Offset: 2

N. J. A. Sloane, Jul 05 2015

nonn

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

Cf. A259701.

Cf. A301620 (essentially twice this sequence).

A000682 = Import["https://oeis.org/A000682/b000682.txt", "Table"][[All, 2]];
a[n_] := If[n <= 2, 0, A000682[[n]]/2 - A000682[[n - 1]]];
a /@ Range[2, 32] (* Jean-François Alcover, Sep 03 2019, from A301620 *)

a(n) = A000682(n)/2 - A000682(n-1) for n > 2.

a(12) from Andrew Howroyd, Dec 07 2018

More terms (using the terms of A301620) from Joerg Arndt, Dec 25 2018

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

Original entry on oeis.org

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

N. J. A. Sloane, Jul 05 2015

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)

			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;
  ...

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

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

Row sums are A000682. First column is A000560.

Cf. A259701.

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

Corrected and extended by Roger Ford, Jul 06 2016

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A259702 Row sums of A259701 except first column.

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

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