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A231210 Number T(n,k) of permutations of [n] with exactly k (possibly overlapping) occurrences of some of the consecutive patterns 123, 1432, 2431, 3421; triangle T(n,k), n>=0, 0<=k<=max(0,n-2), read by rows. 4
1, 1, 2, 5, 1, 14, 9, 1, 46, 59, 14, 1, 177, 358, 164, 20, 1, 790, 2235, 1589, 398, 27, 1, 4024, 14658, 15034, 5659, 909, 35, 1, 23056, 103270, 139465, 77148, 17875, 2021, 44, 1, 146777, 778451, 1334945, 970679, 341071, 52380, 4442, 54, 1, 1027850, 6315499 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Alois P. Heinz, Rows n = 0..142, flattened

A. Baxter, B. Nakamura, and D. Zeilberger, Automatic generation of theorems and proofs on enumerating consecutive Wilf-classes

S. Kitaev and T. Mansour, On multi-avoidance of generalized patterns

EXAMPLE

T(3,1) = 1: 123.

T(4,0) = 14: 1324, 1423, 2143, 2314, 2413, 3142, 3214, 3241, 3412, 4132, 4213, 4231, 4312, 4321.

T(4,1) = 9: 1243, 1342, 1432, 2134, 2341, 2431, 3124, 3421, 4123.

T(4,2) = 1: 1234.

T(5,2) = 14: 12354, 12453, 12543, 13452, 13542, 14532, 21345, 23451, 23541, 24531, 31245, 34521, 41235, 51234.

T(5,3) = 1: 12345.

Triangle T(n,k) begins:

:  0 :      1;

:  1 :      1;

:  2 :      2;

:  3 :      5,      1;

:  4 :     14,      9,       1;

:  5 :     46,     59,      14,      1;

:  6 :    177,    358,     164,     20,      1;

:  7 :    790,   2235,    1589,    398,     27,     1;

:  8 :   4024,  14658,   15034,   5659,    909,    35,    1;

:  9 :  23056, 103270,  139465,  77148,  17875,  2021,   44,  1;

: 10 : 146777, 778451, 1334945, 970679, 341071, 52380, 4442, 54, 1;

MAPLE

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

      add(b(u+j-1, o-j, [2, 2, 2][t])*`if`(t=2, x, 1), j=1..o)+

      add(b(u-j, o+j-1, [1, 3, 1][t])*`if`(t=3, x, 1), j=1..u)))

    end:

T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 0, 1)):

seq(T(n), n=0..14);

MATHEMATICA

b[u_, o_, t_] := b[u, o, t] = If[u+o == 0, 1, Expand[ Sum[b[u+j-1, o-j, {2, 2, 2}[[t]]]*If[t == 2, x, 1], {j, 1, o}] + Sum[b[u-j, o+j-1, {1, 3, 1}[[t]]]*If[t == 3, x, 1], {j, 1, u}]]]; T[n_] := Function[{p}, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, 0, 1]]; Table[T[n], {n, 0, 14}] // Flatten (* Jean-Fran├žois Alcover, Feb 11 2015, after Alois P. Heinz *)

CROSSREFS

Columns k=0-2 give: A231211, A231228, A228422.

Row sums give: A000142.

Cf. A049774, A177479.

Sequence in context: A101282 A263776 A145879 * A178978 A101895 A260670

Adjacent sequences:  A231207 A231208 A231209 * A231211 A231212 A231213

KEYWORD

nonn,tabf

AUTHOR

Alois P. Heinz, Nov 05 2013

STATUS

approved

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Last modified October 5 16:55 EDT 2022. Contains 357259 sequences. (Running on oeis4.)