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A264781 Number T(n,k) of permutations of [n] with exactly k (possibly overlapping) occurrences of the consecutive pattern 45321; triangle T(n,k), n>=0, 0<=k<=max(0,floor((n-1)/4)), read by rows. 5
1, 1, 2, 6, 24, 119, 1, 708, 12, 4914, 126, 38976, 1344, 347765, 15110, 5, 3447712, 180736, 352, 37598286, 2308548, 9966, 447294144, 31481472, 225984, 5764747515, 457520055, 4753185, 45, 80011430240, 7068885600, 97954080, 21280, 1189835682714, 115808906178 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Consecutive patterns 12354, 21345, 54312 give the same triangle.

LINKS

Alois P. Heinz, Rows n = 0..170, flattened

FORMULA

Sum_{k>0} k * T(n,k) = A062199(n-5) for n>4.

EXAMPLE

T(5,1) = 1: 45321.

T(6,1) = 12: 156432, 256431, 356421, 453216, 456321, 463215, 546321, 563214, 564213, 564312, 564321, 645321.

T(9,2) = 5: 786549321, 796548321, 896547321, 897546321, 897645321.

Triangle T(n,k) begins:

00 :           1;

01 :           1;

02 :           2;

03 :           6;

04 :          24;

05 :         119,          1;

06 :         708,         12;

07 :        4914,        126;

08 :       38976,       1344;

09 :      347765,      15110,        5;

10 :     3447712,     180736,      352;

11 :    37598286,    2308548,     9966;

12 :   447294144,   31481472,   225984;

13 :  5764747515,  457520055,  4753185,    45;

14 : 80011430240, 7068885600, 97954080, 21280;

MAPLE

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

       b(u+j-1, o-j, `if`(u+j-3<j, 0, j)), j=1..o)+ expand(

      `if`(t=-2, x, 1)*add(b(u-j, o+j-1, `if`(j<t or t=-2, 0,

      `if`(t>0, -1, `if`(t=-1, -2, 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..17);

CROSSREFS

Columns k=0-1 give: A202213, A264896.

Row sums give A000142.

T(4n+1,n) gives A007696.

Cf. A002265, A007696, A062199.

Sequence in context: A248837 A005394 A095818 * A224316 A256195 A256196

Adjacent sequences:  A264778 A264779 A264780 * A264782 A264783 A264784

KEYWORD

nonn,tabf

AUTHOR

Alois P. Heinz, Nov 24 2015

STATUS

approved

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Last modified June 20 17:43 EDT 2019. Contains 324234 sequences. (Running on oeis4.)