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A259698 Triangle read by rows: T(n,k) = number of permutations without overlaps having k increasing runs. 2
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

2,5

COMMENTS

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.  Ex: 176852943 has 3 runs 123 (left to right), 34567 (right to left), 789 (right to left).

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,

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,

...

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}

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

CROSSREFS

Row sums give A000682.

Cf. A259700.

Sequence in context: A132731 A128966 A055907 * A274643 A172991 A203906

Adjacent sequences:  A259695 A259696 A259697 * A259699 A259700 A259701

KEYWORD

nonn,tabl

AUTHOR

N. J. A. Sloane, Jul 05 2015

EXTENSIONS

Corrected and extended by Roger Ford, Jul 06, 2016.

STATUS

approved

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Last modified September 16 06:38 EDT 2019. Contains 327090 sequences. (Running on oeis4.)