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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

Alois P. Heinz, Rows n = 0..50, flattened

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-Fran├žois Alcover, Jan 16 2017, after Alois P. Heinz *)

CROSSREFS

Columns k=0-10 give: A000110, A264460, A264461, A264462, A264463, A264464, A264465, A264466, A264467, A264468, A264469.

Row sums give A000142.

Cf. A001754, A125811, A260665, A263776.

Sequence in context: A231210 A178978 A101895 * A260665 A110220 A119518

Adjacent sequences:  A260667 A260668 A260669 * A260671 A260672 A260673

KEYWORD

nonn,tabf

AUTHOR

Alois P. Heinz, Nov 14 2015

STATUS

approved

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Last modified April 17 08:34 EDT 2021. Contains 343064 sequences. (Running on oeis4.)