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On multi-avoidance of right angled numbered polyomino patterns. (English) Zbl 1081.05024

Summary: S. Kitaev, T. Mansour and A. Vella [On avoidance of numbered polyomino patterns; http://www.math.chalmers.se/Math/Research/Combinatorics/preprints/kitaev/kmv.ps] introduced numbered polyomino patterns that generalize the concept of pattern avoidance from permutations and words to numbered polyominoes. We study simultaneous avoidance of two or more right-angled numbered polyomino patterns, which are 0-1 labellings of the essentially unique convex two-dimensional polyomino shape with 3 tiles. It turns out that this study gives relations to several combinatorial structures.

MSC:

05B50 Polyominoes
05A15 Exact enumeration problems, generating functions

Keywords:

permutations; words
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Online Encyclopedia of Integer Sequences:

a(0) = 1, thereafter a(n) = 4n.
Multiples of 10: a(n) = 10 * n.
a(n) = 6*n + 4.
a(n) = 10n+2.
a(n) = 12*n + 4.
a(n) = 3*2^n - 2.
a(n) = 7 * 2^n - 6.
Dimension of the space of weight 2n cusp forms for Gamma_0( 62 ).
Number of 4 X n 0-1 matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (00;1), (10;0) and (11;0).
Number of 3 X n 0-1 matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (00;1), (10;0) and (10;1).
Number of 5 X n 0-1 matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (00;1), (01;0), (10;0) and (11;0).
Number of 5 X n binary matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (00;1), (01,1) and (11;0).
Number of 3 X n binary matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (00;1) and (11;0).
Number of 4 X n binary matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (00;1) and (11;0).
Number of 3 X n binary matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (10;0) and (01;1).
Number of 4 X n binary matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (10;0) and (01;1).
Number of 2 X n 0-1 matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (00;1), (01;0), (10;0) and (01;1).
Number of 3 X n 0-1 matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (00;1), (01;0), (10;0) and (01;1).
Number of 4 X n 0-1 matrices avoiding simultaneously the right angled numbered polyomino patterns (ranpp) (00;1), (01;0), (10;0) and (01;1).
The number of permutations avoiding simultaneously consecutive patterns 213 and 231.