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%I #29 Dec 07 2018 16:43:15
%S 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,
%T 98,90,32,21,6,4,1,703,278,243,94,54,21,8,4,1,2105,812,745,270,175,57,
%U 32,8,5,1,6099,2385,2108,808,485,181,84,32,10,5,1
%N Triangle read by rows: T(n,k) = number of permutations without overlaps in which the first increasing run has length k.
%C 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.
%C From _Roger Ford_, Jul 06 2016: (Start)
%C To determine the first increasing run of the permutation 176852943 start on the left and move to the right counting the consecutive integers.
%C (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)
%D A. Sade, Sur les Chevauchements des Permutations, published by the author, Marseille, 1949
%H Albert Sade, <a href="/A000108/a000108_17.pdf">Sur les Chevauchements des Permutations</a>, published by the author, Marseille, 1949. [Annotated scanned copy]
%e Triangle begins:
%e 1;
%e 1, 1;
%e 2, 1, 1;
%e 5, 2, 2, 1;
%e 12, 5, 4, 2, 1;
%e 33, 13, 12, 4, 3, 1;
%e 87, 35, 30, 12, 6, 3, 1;
%e 252, 98, 90, 32, 21, 6, 4, 1;
%e 703, 278, 243, 94, 54, 21, 8, 4, 1;
%e 2105, 812, 745, 270, 175, 57, 32, 8, 5, 1;
%e 6099, 2385, 2108, 808, 485, 181, 84, 32, 10, 5, 1;
%e ...
%o (PARI)
%o Overlapfree(v)={for(i=1, #v, for(j=i+1, v[i]-1, if(v[j]>v[i], return(0)))); 1}
%o 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}
%o FirstRunLen(v)={my(e=1); for(i=1, #v, if(v[i]==e, e++)); e-2}
%o 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}
%o for(n=2, 8, print(row(n))) \\ _Andrew Howroyd_, Dec 07 2018
%Y Row sums are A000682. First column is A000560.
%Y Cf. A259701.
%K nonn,tabl
%O 2,4
%A _N. J. A. Sloane_, Jul 05 2015
%E Corrected and extended by _Roger Ford_, Jul 06 2016
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