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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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Pattern 2143 gives the same triangle.

LINKS

Alois P. Heinz, Rows n = 0..140, flattened

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-Fran├žois Alcover, Jan 16 2017, translated from Maple_ *)

CROSSREFS

Columns k=0-10 give: A113229, A264320, A264321, A264322, A264323, A264324, A264325, A264326, A264327, A264328, A264329.

Row sums give A000142.

Cf. A004526, A061206, A264173 (pattern 1324).

Sequence in context: A060803 A213134 A140837 * A264173 A220183 A177252

Adjacent sequences:  A264316 A264317 A264318 * A264320 A264321 A264322

KEYWORD

nonn,tabf

AUTHOR

Alois P. Heinz, Nov 11 2015

STATUS

approved

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Last modified September 19 05:33 EDT 2020. Contains 337176 sequences. (Running on oeis4.)