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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.

%I A259701 #12 Dec 07 2018 16:42:55
%S A259701 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,
%T A259701 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,
%U A259701 6099,280,1296,63,215,6,27,0,2,0,0
%N 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.
%C A259701 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).
%D A259701 A. Sade, Sur les Chevauchements des Permutations, published by the author, Marseille, 1949
%H A259701 Albert Sade, <a href="/A000108/a000108_17.pdf">Sur les Chevauchements des Permutations</a>, published by the author, Marseille, 1949. [Annotated scanned copy]
%e A259701 Triangle begins:
%e A259701      0;
%e A259701      1,   0;
%e A259701      2,   0,   0;
%e A259701      5,   0,   1,   0;
%e A259701     12,   0,   2,   0,  0;
%e A259701     33,   1,   7,   0,  1, 0;
%e A259701     87,   2,  17,   0,  2, 0,  0;
%e A259701    252,  11,  55,   2,  9, 0,  1, 0;
%e A259701    703,  26, 145,   4, 22, 0,  2, 0, 0;
%e A259701   2105, 109, 467,  27, 81, 3, 11, 0, 1, 0;
%e A259701   ...
%o A259701 (PARI)
%o A259701 Overlapfree(v)={for(i=1, #v, for(j=i+1, v[i]-1, if(v[j]>v[i], return(0)))); 1}
%o A259701 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 A259701 FirstRunLen(v)={my(e=1); for(i=1, #v, if(v[i]==e, e++)); e-2}
%o A259701 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}
%o A259701 for(n=2, 8, print(row(n))) \\ _Andrew Howroyd_, Dec 07 2018
%Y A259701 Row sums excluding the first column give A259702.
%Y A259701 First column is A000560.
%Y A259701 Cf. A259703.
%K A259701 nonn,tabl,more
%O A259701 2,4
%A A259701 _N. J. A. Sloane_, Jul 05 2015
%E A259701 a(49) corrected and a(57)-a(67) from _Andrew Howroyd_, Dec 07 2018