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A188919 Triangle read by rows: T(n,k) = number of permutations of length n with k inversions that avoid the "dashed pattern" 1-32.

%I A188919 #33 Dec 28 2018 15:33:36
%S A188919 1,1,1,1,1,1,2,1,1,1,2,4,3,3,1,1,1,2,4,7,8,9,9,6,4,1,1,1,2,4,7,13,16,
%T A188919 22,26,29,26,23,17,10,5,1,1,1,2,4,7,13,22,31,44,60,74,89,95,98,93,82,
%U A188919 63,47,29,15,6,1,1,1,2,4,7,13,22,38,55,83,116,160,207,259,304,347,375,386,378,348,304,249,190,131,85,46,21,7,1
%N A188919 Triangle read by rows: T(n,k) = number of permutations of length n with k inversions that avoid the "dashed pattern" 1-32.
%C A188919 Row n has length 1 + binomial(n,2) and sum A000110(n) (a Bell number).
%H A188919 Alois P. Heinz, <a href="/A188919/b188919.txt">Rows n = 0..50, flattened</a>
%H A188919 A. M. Baxter, <a href="https://pdfs.semanticscholar.org/2c5d/79e361d3aecb25c380402144177ad7cd9dc8.pdf">Algorithms for Permutation Statistics</a>, Ph. D. Dissertation, Rutgers University, May 2011.
%H A188919 Andrew Baxter, <a href="/A188919/a188919.txt">Additional terms, formatted as a table.</a>
%H A188919 Andrew M. Baxter and Lara K. Pudwell, <a href="http://arxiv.org/abs/1108.2642">Enumeration schemes for dashed patterns</a>, arXiv preprint arXiv:1108.2642, 2011
%H A188919 Jean-Christophe Novelli, Jean-Yves Thibon, Frédéric Toumazet, <a href="https://arxiv.org/abs/1705.08113">Noncommutative Bell polynomials and the dual immaculate basis</a>, arXiv:1705.08113 [math.CO], 2017.
%e A188919 Triangle begins:
%e A188919 1
%e A188919 1
%e A188919 1 1
%e A188919 1 1 2 1
%e A188919 1 1 2 4 3 3 1
%e A188919 1 1 2 4 7 8 9 9 6 4 1
%e A188919 ...
%p A188919 b:= proc(u, o) option remember; expand(`if`(u+o=0, 1,
%p A188919        add(b(u-j, o+j-1)*x^(o+j-1), j=1..u)+
%p A188919        add(`if`(u=0, b(u+j-1, o-j)*x^(o-j), 0), j=1..o)))
%p A188919     end:
%p A188919 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(0, n)):
%p A188919 seq(T(n), n=0..10);  # _Alois P. Heinz_, Nov 14 2015
%t A188919 b[u_, o_] := b[u, o] = Expand[If[u+o == 0, 1, Sum[b[u-j, o+j-1]* x^(o+j-1), {j, 1, u}] + Sum[If[u == 0, b[u+j-1, o-j]*x^(o-j), 0], {j, 1, o}]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}] ][b[0, n]]; Table[T[n], {n, 0, 10}] // Flatten (* _Jean-François Alcover_, Feb 01 2016, after _Alois P. Heinz_ *)
%Y A188919 The column limits are given by A188920.
%Y A188919 Cf. A000110, A161680.
%K A188919 nonn,tabf
%O A188919 0,7
%A A188919 _N. J. A. Sloane_, Apr 13 2011
%E A188919 More terms from _Andrew Baxter_, May 17 2011.