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A260670 Number T(n,k) of permutations of [n] with exactly k (possibly overlapping) occurrences of the generalized pattern 23-1; triangle T(n,k), n>=0, 0<=k<=A125811(n)-1, read by rows. 13
1, 1, 2, 5, 1, 15, 6, 3, 52, 32, 23, 10, 3, 203, 171, 152, 98, 62, 22, 11, 1, 877, 944, 984, 791, 624, 392, 240, 111, 55, 18, 4, 4140, 5444, 6460, 6082, 5513, 4302, 3328, 2141, 1393, 780, 432, 187, 88, 24, 6, 21147, 32919, 43626, 46508, 46880, 41979, 36774 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,3
COMMENTS
Patterns 1-32, 3-12, 21-3 give the same sequence.
LINKS
A. Claesson and T. Mansour, Counting occurrences of a pattern of type (1,2) or (2,1) in permutations, arXiv:math/0110036 [math.CO], 2001
FORMULA
Sum_{k>0} k * T(n,k) = A001754(n).
EXAMPLE
T(3,1) = 1: 231.
T(4,1) = 6: 1342, 2314, 2413, 2431, 3241, 4231.
T(4,2) = 3: 2341, 3412, 3421.
T(5,2) = 23: 13452, 14523, 14532, 23415, 23514, 23541, 24351, 25341, 32451, 34125, 34152, 34215, 35124, 35142, 35214, 35412, 35421, 42351, 43512, 43521, 52341, 53412, 53421.
T(5,3) = 10: 23451, 24513, 24531, 34251, 35241, 45123, 45132, 45213, 45312, 45321.
T(5,4) = 3: 34512, 34521, 45231.
Triangle T(n,k) begins:
0 : 1;
1 : 1;
2 : 2;
3 : 5, 1;
4 : 15, 6, 3;
5 : 52, 32, 23, 10, 3;
6 : 203, 171, 152, 98, 62, 22, 11, 1;
7 : 877, 944, 984, 791, 624, 392, 240, 111, 55, 18, 4;
MAPLE
b:= proc(u, o) option remember;
`if`(u+o=0, 1, add(b(u-j, o+j-1), j=1..u)+
add(expand(b(u+j-1, o-j)*x^u), j=1..o))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0)):
seq(T(n), n=0..10);
MATHEMATICA
b[u_, o_] := b[u, o] = If[u+o == 0, 1, Sum[b[u-j, o+j-1], {j, 1, u}] + Sum[Expand[b[u+j-1, o-j]*x^u], {j, 1, o}]]; T[n_] := Function[p, Table[ Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, 0]]; Table[T[n], {n, 0, 10}] // Flatten (* Jean-François Alcover, Jan 16 2017, after Alois P. Heinz *)
CROSSREFS
Row sums give A000142.
Sequence in context: A231210 A178978 A101895 * A260665 A110220 A349106
KEYWORD
nonn,tabf
AUTHOR
Alois P. Heinz, Nov 14 2015
STATUS
approved

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Last modified June 23 01:26 EDT 2024. Contains 373629 sequences. (Running on oeis4.)