%I
%S 1,1,0,1,2,0,1,8,2,0,1,22,26,0,0,1,52,168,42,0,0,1,114,804,692,42,0,0,
%T 1,240,3270,6500,1866,0,0,0,1,494,12054,46304,34078,3060,0,0,0,1,1004,
%U 41708,279566,413878,122830,3060,0,0,0,1,2026,138320,1514324
%N Triangle read by rows: T(n,k) is the number of permutations of length n composed of exactly k overlapping adjacent runs (for n >= 1 and 1 <= k <= n).
%C Permutations of A307030 grouped by number of runs. Thus row sums give A307030.
%C Each column admits a rational generating function (Asinowski et al.).
%H Bjarki Ágúst Guðmundsson, <a href="/A309993/b309993.txt">Table of n, a(n) for n = 1..5050</a>
%H Andrei Asinowski, Cyril Banderier, Sara Billey, Benjamin Hackl, Svante Linusson, <a href="https://lipn.fr/~cb/Papers/popstack.pdf">Popstack sorting and its image: Permutations with overlapping runs</a> (2019), preprint.
%H Anders Claesson, Bjarki Ágúst Guðmundsson, Jay Pantone, <a href="https://arxiv.org/abs/1908.08910">Counting popstacked permutations in polynomial time</a>, arXiv:1908.08910 [math.CO], 2019.
%F G.f. for column k=1: x/(1x).
%F G.f. for column k=2: 2*x^3/((1x)^2*(12*x)).
%F G.f. for column k=3: 2*x^4*(6*x^2  3*x  1)/((1x)^3*(12*x)^2*(13*x)).
%F G.f. for column k=4: 2*x^6*(144*x^4  180*x^3  5*x^2 + 74*x  21)/((1x)^4*(12*x)^3*(13*x)^2*(14*x)).
%F G.f. for column k=5: 2*x^7*(17280*x^8  37600*x^7 + 12784*x^6 + 33060*x^5  40581*x^4 + 18982*x^3  3856*x^2 + 198*x + 21)/((1x)^5*(12*x)^4*(13*x)^3*(14*x)^2*(15*x)).
%e For n = 3 the permutations with overlapping runs are 123, 132, 213. The first has k = 1 runs, the other two have k = 2 runs. Thus T(3,1) = 1, T(3,2) = 2, T(3,3) = 0.
%e Triangle begins:
%e 1;
%e 1, 0;
%e 1, 2, 0;
%e 1, 8, 2, 0;
%e 1, 22, 26, 0, 0;
%e 1, 52, 168, 42, 0, 0;
%e 1, 114, 804, 692, 42, 0, 0;
%e 1, 240, 3270, 6500, 1866, 0, 0, 0;
%e 1, 494, 12054, 46304, 34078, 3060, 0, 0, 0;
%e 1, 1004, 41708, 279566, 413878, 122830, 3060, 0, 0, 0;
%e ...
%Y Cf. A307030.
%K nonn,tabl
%O 1,5
%A _Bjarki Ágúst Guðmundsson_, Aug 26 2019
