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A231384 Number T(n,k) of permutations of [n] with exactly k (possibly overlapping) occurrences of some of the consecutive step patterns UUD, UDU, DUU (U=up, D=down); triangle T(n,k), n>=0, 0<=k<=max(0,n-3), read by rows. 6
1, 1, 2, 6, 13, 11, 39, 52, 29, 158, 233, 230, 99, 674, 1344, 1537, 1118, 367, 3304, 8197, 11208, 10200, 5868, 1543, 19511, 49846, 89657, 95624, 67223, 33118, 7901, 122706, 351946, 724755, 907078, 781827, 492285, 206444, 41759, 834131, 2799536, 6010150 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

LINKS

Alois P. Heinz, Rows n = 0..80, 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(4,1) = 11: 1243, 1342, 2341 (UUD), 1324, 1423, 2314, 2413, 3412 (UDU), 2134, 3124, 4123 (DUU).

T(5,0) = 39: 12345, 14325, 15324, ..., 54231, 54312, 54321.

T(5,1) = 52: 12354, 12453, 12543, ..., 53124, 53412, 54123.

T(5,2) = 29: 12435, 12534, 13245, ..., 51243, 51342, 52341.

Triangle T(n,k) begins:

: 0 :     1;

: 1 :     1;

: 2 :     2;

: 3 :     6;

: 4 :    13,    11;

: 5 :    39,    52,    29;

: 6 :   158,   233,   230,    99;

: 7 :   674,  1344,  1537,  1118,   367;

: 8 :  3304,  8197, 11208, 10200,  5868,  1543;

: 9 : 19511, 49846, 89657, 95624, 67223, 33118, 7901;

MAPLE

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

     add(b(u+j-1, o-j, [2, 3, 3, 6, 6, 3][t])*

             `if`(t in [5, 6], x, 1), j=1..o)+

     add(b(u-j, o+j-1, [4, 5, 5, 4, 4, 5][t])*

             `if`(t=3, x, 1), j=1..u)))

    end:

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

                     (add(b(j-1, n-j, 1), j=1..n))):

seq(T(n), n=0..12);

MATHEMATICA

b[u_, o_, t_] := b[u, o, t] = If[u+o == 0, 1, Expand[Sum[b[u+j-1, o-j, {2, 3, 3, 6, 6, 3}[[t]]]*If[t == 5 || t == 6, x, 1], {j, 1, o}] + Sum[b[u-j, o+j-1, {4, 5, 5, 4, 4, 5}[[t]]]*If[t == 3, x, 1], {j, 1, u}]]]; T[n_] := If[n == 0, 1, Function[{p},  Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][Sum[b[j-1, n-j, 1], {j, 1, n}]]]; Table[T[n], {n, 0, 12}] // Flatten (* Jean-Fran├žois Alcover, Feb 05 2015, after Alois P. Heinz *)

CROSSREFS

Columns k=0-2 give: A231385, A231386, A228408.

Diagonal gives: A231410.

Row sums give: A000142.

Cf. A295987.

Sequence in context: A226603 A116534 A130533 * A082722 A030416 A277690

Adjacent sequences:  A231381 A231382 A231383 * A231385 A231386 A231387

KEYWORD

nonn,tabf

AUTHOR

Alois P. Heinz, Nov 08 2013

STATUS

approved

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Last modified October 15 19:25 EDT 2019. Contains 328037 sequences. (Running on oeis4.)