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A231210 Number T(n,k) of permutations of [n] with exactly k (possibly overlapping) occurrences of some of the consecutive patterns 123, 1432, 2431, 3421; triangle T(n,k), n>=0, 0<=k<=max(0,n-2), read by rows.

%I A231210 #30 Feb 11 2015 10:46:51
%S A231210 1,1,2,5,1,14,9,1,46,59,14,1,177,358,164,20,1,790,2235,1589,398,27,1,
%T A231210 4024,14658,15034,5659,909,35,1,23056,103270,139465,77148,17875,2021,
%U A231210 44,1,146777,778451,1334945,970679,341071,52380,4442,54,1,1027850,6315499
%N A231210 Number T(n,k) of permutations of [n] with exactly k (possibly overlapping) occurrences of some of the consecutive patterns 123, 1432, 2431, 3421; triangle T(n,k), n>=0, 0<=k<=max(0,n-2), read by rows.
%H A231210 Alois P. Heinz, <a href="/A231210/b231210.txt">Rows n = 0..142, flattened</a>
%H A231210 A. Baxter, B. Nakamura, and D. Zeilberger, <a href="http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/auto.html">Automatic generation of theorems and proofs on enumerating consecutive Wilf-classes</a>
%H A231210 S. Kitaev and T. Mansour, <a href="http://www.ru.is/kennarar/sergey/index_files/Papers/multi_avoid_gen_patterns.pdf">On multi-avoidance of generalized patterns</a>
%e A231210 T(3,1) = 1: 123.
%e A231210 T(4,0) = 14: 1324, 1423, 2143, 2314, 2413, 3142, 3214, 3241, 3412, 4132, 4213, 4231, 4312, 4321.
%e A231210 T(4,1) = 9: 1243, 1342, 1432, 2134, 2341, 2431, 3124, 3421, 4123.
%e A231210 T(4,2) = 1: 1234.
%e A231210 T(5,2) = 14: 12354, 12453, 12543, 13452, 13542, 14532, 21345, 23451, 23541, 24531, 31245, 34521, 41235, 51234.
%e A231210 T(5,3) = 1: 12345.
%e A231210 Triangle T(n,k) begins:
%e A231210 :  0 :      1;
%e A231210 :  1 :      1;
%e A231210 :  2 :      2;
%e A231210 :  3 :      5,      1;
%e A231210 :  4 :     14,      9,       1;
%e A231210 :  5 :     46,     59,      14,      1;
%e A231210 :  6 :    177,    358,     164,     20,      1;
%e A231210 :  7 :    790,   2235,    1589,    398,     27,     1;
%e A231210 :  8 :   4024,  14658,   15034,   5659,    909,    35,    1;
%e A231210 :  9 :  23056, 103270,  139465,  77148,  17875,  2021,   44,  1;
%e A231210 : 10 : 146777, 778451, 1334945, 970679, 341071, 52380, 4442, 54, 1;
%p A231210 b:= proc(u, o, t) option remember; `if`(u+o=0, 1, expand(
%p A231210       add(b(u+j-1, o-j, [2, 2, 2][t])*`if`(t=2, x, 1), j=1..o)+
%p A231210       add(b(u-j, o+j-1, [1, 3, 1][t])*`if`(t=3, x, 1), j=1..u)))
%p A231210     end:
%p A231210 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0, 1)):
%p A231210 seq(T(n), n=0..14);
%t A231210 b[u_, o_, t_] := b[u, o, t] = If[u+o == 0, 1, Expand[ Sum[b[u+j-1, o-j, {2, 2, 2}[[t]]]*If[t == 2, x, 1], {j, 1, o}] + Sum[b[u-j, o+j-1, {1, 3, 1}[[t]]]*If[t == 3, x, 1], {j, 1, u}]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, 0, 1]]; Table[T[n], {n, 0, 14}] // Flatten (* _Jean-François Alcover_, Feb 11 2015, after _Alois P. Heinz_ *)
%Y A231210 Columns k=0-2 give: A231211, A231228, A228422.
%Y A231210 Row sums give: A000142.
%Y A231210 Cf. A049774, A177479.
%K A231210 nonn,tabf
%O A231210 0,3
%A A231210 _Alois P. Heinz_, Nov 05 2013