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A309993 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).

%I A309993 #21 Aug 29 2019 20:45:28
%S A309993 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 A309993 1,240,3270,6500,1866,0,0,0,1,494,12054,46304,34078,3060,0,0,0,1,1004,
%U A309993 41708,279566,413878,122830,3060,0,0,0,1,2026,138320,1514324
%N A309993 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 A309993 Permutations of A307030 grouped by number of runs. Thus row sums give A307030.
%C A309993 Each column admits a rational generating function (Asinowski et al.).
%H A309993 Bjarki Ágúst Guðmundsson, <a href="/A309993/b309993.txt">Table of n, a(n) for n = 1..5050</a>
%H A309993 Andrei Asinowski, Cyril Banderier, Sara Billey, Benjamin Hackl, Svante Linusson, <a href="https://lipn.fr/~cb/Papers/popstack.pdf">Pop-stack sorting and its image: Permutations with overlapping runs</a> (2019), preprint.
%H A309993 Anders Claesson, Bjarki Ágúst Guðmundsson, Jay Pantone, <a href="https://arxiv.org/abs/1908.08910">Counting pop-stacked permutations in polynomial time</a>, arXiv:1908.08910 [math.CO], 2019.
%F A309993 G.f. for column k=1: x/(1-x).
%F A309993 G.f. for column k=2: 2*x^3/((1-x)^2*(1-2*x)).
%F A309993 G.f. for column k=3: -2*x^4*(6*x^2 - 3*x - 1)/((1-x)^3*(1-2*x)^2*(1-3*x)).
%F A309993 G.f. for column k=4: -2*x^6*(144*x^4 - 180*x^3 - 5*x^2 + 74*x - 21)/((1-x)^4*(1-2*x)^3*(1-3*x)^2*(1-4*x)).
%F A309993 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)/((1-x)^5*(1-2*x)^4*(1-3*x)^3*(1-4*x)^2*(1-5*x)).
%e A309993 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 A309993 Triangle begins:
%e A309993   1;
%e A309993   1,    0;
%e A309993   1,    2,     0;
%e A309993   1,    8,     2,      0;
%e A309993   1,   22,    26,      0,      0;
%e A309993   1,   52,   168,     42,      0,      0;
%e A309993   1,  114,   804,    692,     42,      0,    0;
%e A309993   1,  240,  3270,   6500,   1866,      0,    0, 0;
%e A309993   1,  494, 12054,  46304,  34078,   3060,    0, 0, 0;
%e A309993   1, 1004, 41708, 279566, 413878, 122830, 3060, 0, 0, 0;
%e A309993   ...
%Y A309993 Cf. A307030.
%K A309993 nonn,tabl
%O A309993 1,5
%A A309993 _Bjarki Ágúst Guðmundsson_, Aug 26 2019