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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.
13
1, 1, 2, 6, 23, 1, 110, 10, 631, 88, 1, 4223, 794, 23, 32301, 7639, 379, 1, 277962, 79164, 5706, 48, 2657797, 885128, 84354, 1520, 1, 27954521, 10657588, 1266150, 38452, 89, 320752991, 137752283, 19621124, 869740, 5461, 1, 3987045780, 1904555934, 316459848
OFFSET
0,3
COMMENTS
Pattern 2143 gives the same triangle.
LINKS
FORMULA
Sum_{k>0} k * T(n,k) = ceiling((n-3)*n!/4!) = A061206(n-3) (for n>3).
EXAMPLE
T(4,1) = 1: 3412.
T(5,1) = 10: 14523, 24513, 34125, 34512, 35124, 43512, 45123, 45132, 45231, 53412.
T(6,2) = 1: 563412.
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.
T(8,3) = 1: 78563412.
T(9,3) = 48: 189674523, 289674513, 389674512, ..., 896745132, 896745231, 978563412.
Triangle T(n,k) begins:
00 : 1;
01 : 1;
02 : 2;
03 : 6;
04 : 23, 1;
05 : 110, 10;
06 : 631, 88, 1;
07 : 4223, 794, 23;
08 : 32301, 7639, 379, 1;
09 : 277962, 79164, 5706, 48;
10 : 2657797, 885128, 84354, 1520, 1;
MAPLE
b:= proc(u, o, t) option remember; `if`(u+o=0, 1, add(expand(
b(u+j-1, o-j, j)*`if`(t<0 and j<1-t, x, 1)), j=1..o)+
add(b(u-j, o+j-1, `if`(t>0 and j>t, t-j, 0)), j=1..u))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0$2)):
seq(T(n), n=0..14);
MATHEMATICA
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_ *)
CROSSREFS
Row sums give A000142.
Cf. A004526, A061206, A264173 (pattern 1324).
Sequence in context: A342865 A350274 A350273 * A264173 A220183 A177252
KEYWORD
nonn,tabf
AUTHOR
Alois P. Heinz, Nov 11 2015
STATUS
approved