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A264319 Number T(n,k) of permutations of [n] with exactly k (possibly overlapping) occurrences of the consecutive pattern 3412; triangle T(n,k), n>=0, 0<=k<=max(0,floor(n/2-1)), read by rows.

%I A264319 #14 Jan 16 2017 08:53:54
%S A264319 1,1,2,6,23,1,110,10,631,88,1,4223,794,23,32301,7639,379,1,277962,
%T A264319 79164,5706,48,2657797,885128,84354,1520,1,27954521,10657588,1266150,
%U A264319 38452,89,320752991,137752283,19621124,869740,5461,1,3987045780,1904555934,316459848
%N A264319 Number T(n,k) of permutations of [n] with exactly k (possibly overlapping) occurrences of the consecutive pattern 3412; triangle T(n,k), n>=0, 0<=k<=max(0,floor(n/2-1)), read by rows.
%C A264319 Pattern 2143 gives the same triangle.
%H A264319 Alois P. Heinz, <a href="/A264319/b264319.txt">Rows n = 0..140, flattened</a>
%F A264319 Sum_{k>0} k * T(n,k) = ceiling((n-3)*n!/4!) = A061206(n-3) (for n>3).
%e A264319 T(4,1) = 1: 3412.
%e A264319 T(5,1) = 10: 14523, 24513, 34125, 34512, 35124, 43512, 45123, 45132, 45231, 53412.
%e A264319 T(6,2) = 1: 563412.
%e A264319 T(7,2) = 23: 1674523, 2674513, 3674512, 4673512, 5614723, 5624713, 5634127, 5634712, 5673412, 5714623, 5724613, 5734126, 5734612, 6573412, 6714523, 6724513, 6734125, 6734512, 6735124, 6745123, 6745132, 6745231, 7563412.
%e A264319 T(8,3) = 1: 78563412.
%e A264319 T(9,3) = 48: 189674523, 289674513, 389674512, ..., 896745132, 896745231, 978563412.
%e A264319 Triangle T(n,k) begins:
%e A264319 00 :       1;
%e A264319 01 :       1;
%e A264319 02 :       2;
%e A264319 03 :       6;
%e A264319 04 :      23,      1;
%e A264319 05 :     110,     10;
%e A264319 06 :     631,     88,     1;
%e A264319 07 :    4223,    794,    23;
%e A264319 08 :   32301,   7639,   379,    1;
%e A264319 09 :  277962,  79164,  5706,   48;
%e A264319 10 : 2657797, 885128, 84354, 1520, 1;
%p A264319 b:= proc(u, o, t) option remember; `if`(u+o=0, 1, add(expand(
%p A264319        b(u+j-1, o-j, j)*`if`(t<0 and j<1-t, x, 1)), j=1..o)+
%p A264319       add(b(u-j, o+j-1, `if`(t>0 and j>t, t-j, 0)), j=1..u))
%p A264319     end:
%p A264319 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0$2)):
%p A264319 seq(T(n), n=0..14);
%t A264319 b[u_, o_, t_] := b[u, o, t] = If[u+o == 0, 1, Sum[Expand[b[u+j-1, o-j, j]*If[t<0 && j<1-t, x, 1]], {j, 1, o}] + Sum[b[u-j, o+j-1, If[t>0 && j>t, t-j, 0]], {j, 1, u}]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, 0, 0]]; Table[T[n], {n, 0, 14}] // Flatten (* _Jean-François Alcover_, Jan 16 2017, translated from Maple_ *)
%Y A264319 Columns k=0-10 give: A113229, A264320, A264321, A264322, A264323, A264324, A264325, A264326, A264327, A264328, A264329.
%Y A264319 Row sums give A000142.
%Y A264319 Cf. A004526, A061206, A264173 (pattern 1324).
%K A264319 nonn,tabf
%O A264319 0,3
%A A264319 _Alois P. Heinz_, Nov 11 2015