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A231210
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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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4
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1, 1, 2, 5, 1, 14, 9, 1, 46, 59, 14, 1, 177, 358, 164, 20, 1, 790, 2235, 1589, 398, 27, 1, 4024, 14658, 15034, 5659, 909, 35, 1, 23056, 103270, 139465, 77148, 17875, 2021, 44, 1, 146777, 778451, 1334945, 970679, 341071, 52380, 4442, 54, 1, 1027850, 6315499
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OFFSET
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0,3
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LINKS
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EXAMPLE
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T(3,1) = 1: 123.
T(4,0) = 14: 1324, 1423, 2143, 2314, 2413, 3142, 3214, 3241, 3412, 4132, 4213, 4231, 4312, 4321.
T(4,1) = 9: 1243, 1342, 1432, 2134, 2341, 2431, 3124, 3421, 4123.
T(4,2) = 1: 1234.
T(5,2) = 14: 12354, 12453, 12543, 13452, 13542, 14532, 21345, 23451, 23541, 24531, 31245, 34521, 41235, 51234.
T(5,3) = 1: 12345.
Triangle T(n,k) begins:
: 0 : 1;
: 1 : 1;
: 2 : 2;
: 3 : 5, 1;
: 4 : 14, 9, 1;
: 5 : 46, 59, 14, 1;
: 6 : 177, 358, 164, 20, 1;
: 7 : 790, 2235, 1589, 398, 27, 1;
: 8 : 4024, 14658, 15034, 5659, 909, 35, 1;
: 9 : 23056, 103270, 139465, 77148, 17875, 2021, 44, 1;
: 10 : 146777, 778451, 1334945, 970679, 341071, 52380, 4442, 54, 1;
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MAPLE
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b:= proc(u, o, t) option remember; `if`(u+o=0, 1, expand(
add(b(u+j-1, o-j, [2, 2, 2][t])*`if`(t=2, x, 1), j=1..o)+
add(b(u-j, o+j-1, [1, 3, 1][t])*`if`(t=3, x, 1), j=1..u)))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0, 1)):
seq(T(n), n=0..14);
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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