A259702 -id:A259702 - cp's OEIS frontend

A301620 a(n) is the total number of top arches with exactly one covering arch for semi-meanders with n top arches.

0, 0, 2, 4, 18, 42, 156, 398, 1398, 3778, 12982, 36522, 124290, 360182, 1220440, 3618090, 12237698, 36938158, 124880222, 382471606, 1293363816, 4009185912, 13565790984, 42478788432, 143851766298, 454339269482, 1539997455570, 4900091676662, 16624834778474, 53240459608298

Offset: 1

Roger Ford, Mar 24 2018

nonn

For n>2, a(n-2) is the number of ways to fold a strip of n stamps with leaf 1 on top and the n leaf not adjacent to the n-1 leaf. Example n = 6, a(6-2) = 4: 125436, 126345, 154362, 163452. - Roger Ford, Mar 29 2019

For n>2, a(n-2) is the number of ways to fold a strip of n stamps with leaf 1 on top and leaf 2 not in the second position and not in the n-th position. Example, for n = 6, a(6-2) = 4: 143265, 156234, 165234, 143256. - Roger Ford, Mar 12 2021

			For n = 4, a(4) = 4.  + + are underneath the starting and ending of each arch with exactly one covering arch.
          /\                  /\
         //\\         /\     //\\       /\
      /\///\\\,  /\/\//\\,  ///\\\/\,  //\\/\/\ .
         +  +         ++     +  +       ++

Jean-François Alcover, Table of n, a(n) for n = 1..43

Cf. A000682, A259689.

A000682 = Import["https://oeis.org/A000682/b000682.txt", "Table"][[All, 2]];
a[n_] := A000682[[n + 2]] - 2*A000682[[n + 1]];
Array[a, 30] (* Jean-François Alcover, Sep 02 2019 *)

a(n) = A000682(n+2) - 2*A000682(n+1).
a(n) = Sum_{k=3..floor((n+3)/2)} (A259689(n+1,k)*(k-2)). - Roger Ford, Dec 10 2018
a(n) = 2*A259702(n+2). - Roger Ford, Dec 24 2018

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.

Original entry on oeis.org

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

N. J. A. Sloane, Jul 05 2015

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).

			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;
  ...

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 excluding the first column give A259702.
First column is A000560.
Cf. A259703.

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

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

cp's OEIS Frontend

A301620 a(n) is the total number of top arches with exactly one covering arch for semi-meanders with n top arches.

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