A259700 -id:A259700 - cp's OEIS frontend

A259699 Erroneous version of A259700.

1, 3, 8, 25, 72, 229, 689, 2222, 6873

Offset: 2

dead

Included in accordance with OEIS policy of including published but incorrect sequences to serve as pointers to the correct versions.

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

A259698 Triangle read by rows: T(n,k) = number of permutations without overlaps having k increasing runs.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 6, 10, 6, 1, 1, 10, 23, 22, 9, 1, 1, 14, 44, 61, 41, 12, 1, 1, 22, 87, 158, 148, 71, 16, 1, 1, 30, 151, 352, 436, 301, 114, 20, 1, 1, 46, 280, 791, 1210, 1092, 589, 175, 25, 1, 1, 62, 464, 1592, 2969, 3317, 2408, 1038, 256, 30, 1

Offset: 2

N. J. A. Sloane, Jul 05 2015

The sums s(n) = Sum_k k*T(n,k) give A259700.

Albert Sade in Sur les Chevauchements des Permutation (published by the author in French in 1949) gave the following example for determining the number of increasing runs in a permutation: 176852943 has 3 runs: 123 (left to right), 34567 (right to left), 789 (right to left).

			Triangle begins:
  1;
  1,  1;
  1,  2,   1;
  1,  4,   4,    1;
  1,  6,  10,    6,    1;
  1, 10,  23,   22,    9,    1;
  1, 14,  44,   61,   41,   12,    1;
  1, 22,  87,  158,  148,   71,   16,    1;
  1, 30, 151,  352,  436,  301,  114,   20,   1;
  1, 46, 280,  791, 1210, 1092,  589,  175,  25,  1;
  1, 62, 464, 1592, 2969, 3377, 2408, 1038, 256, 30, 1;
  ...

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

Row sums give A000682.

Cf. A259700.

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}
Runs(v)={my(u=vector(#v), s=1); for(i=1, #v, u[v[i]]=i); for(i=2, #u-1, if(sign(u[i]-u[i-1])==sign(u[i]-u[i+1]), s++)); s}
row(n)={my(r=vector(n-1)); if(n>=2, forperm(n, v, if(v[1]<>1, break); if(Overlapfree(Chords(v)), r[Runs(v)]++))); r}
for(n=2, 8, print(row(n))) \\ Andrew Howroyd, Dec 07 2018

Corrected and extended by Roger Ford, Jul 06 2016

cp's OEIS Frontend

A259699 Erroneous version of A259700.

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A259698 Triangle read by rows: T(n,k) = number of permutations without overlaps having k increasing runs.

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