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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.
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%I #30 Feb 11 2015 10:46:51

%S 1,1,2,5,1,14,9,1,46,59,14,1,177,358,164,20,1,790,2235,1589,398,27,1,

%T 4024,14658,15034,5659,909,35,1,23056,103270,139465,77148,17875,2021,

%U 44,1,146777,778451,1334945,970679,341071,52380,4442,54,1,1027850,6315499

%N 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 Alois P. Heinz, <a href="/A231210/b231210.txt">Rows n = 0..142, flattened</a>

%H 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 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 T(3,1) = 1: 123.

%e T(4,0) = 14: 1324, 1423, 2143, 2314, 2413, 3142, 3214, 3241, 3412, 4132, 4213, 4231, 4312, 4321.

%e T(4,1) = 9: 1243, 1342, 1432, 2134, 2341, 2431, 3124, 3421, 4123.

%e T(4,2) = 1: 1234.

%e T(5,2) = 14: 12354, 12453, 12543, 13452, 13542, 14532, 21345, 23451, 23541, 24531, 31245, 34521, 41235, 51234.

%e T(5,3) = 1: 12345.

%e Triangle T(n,k) begins:

%e : 0 : 1;

%e : 1 : 1;

%e : 2 : 2;

%e : 3 : 5, 1;

%e : 4 : 14, 9, 1;

%e : 5 : 46, 59, 14, 1;

%e : 6 : 177, 358, 164, 20, 1;

%e : 7 : 790, 2235, 1589, 398, 27, 1;

%e : 8 : 4024, 14658, 15034, 5659, 909, 35, 1;

%e : 9 : 23056, 103270, 139465, 77148, 17875, 2021, 44, 1;

%e : 10 : 146777, 778451, 1334945, 970679, 341071, 52380, 4442, 54, 1;

%p b:= proc(u, o, t) option remember; `if`(u+o=0, 1, expand(

%p add(b(u+j-1, o-j, [2, 2, 2][t])*`if`(t=2, x, 1), j=1..o)+

%p add(b(u-j, o+j-1, [1, 3, 1][t])*`if`(t=3, x, 1), j=1..u)))

%p end:

%p T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0, 1)):

%p seq(T(n), n=0..14);

%t 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 Columns k=0-2 give: A231211, A231228, A228422.

%Y Row sums give: A000142.

%Y Cf. A049774, A177479.

%K nonn,tabf

%O 0,3

%A _Alois P. Heinz_, Nov 05 2013