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Pattern-Avoidance and Fuss-Catalan Numbers
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Per Alexandersson

Department of Mathematics

Stockholm University

S-106 91 Stockholm

Sweden

Frether Getachew Kebede

Department of Mathematics

Addis Ababa University

1176 Addis Ababa

Ethiopia

Samuel Asefa Fufa

Department of Mathematics

Addis Ababa University

1176 Addis Ababa

Ethiopia

Dun Qiu

Center for Combinatorics, LPMC

Nankai University

Tianjin 300071

P. R. China

**Abstract:**

We study a subset of permutations where entries are restricted to having
the same remainder as the index, modulo some integer *k* ≥ 2. We show
that by also imposing the classical 132- or 213-avoidance restriction on
the permutations, we recover the Fussâ€“Catalan numbers and some special
cases of the Raney numbers. Surprisingly, an analogous statement also
holds when we impose the mod *k* restriction on a Catalan family of
subexcedant functions. Finally, we completely enumerate all combinations
of mod-*k*-alternating permutations that avoid two patterns of length
3. This is analogous to the systematic study by Simion and Schmidt,
of permutations avoiding two patterns of length 3.

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(Concerned with sequences
A000027
A000045
A000079
A000124
A047749
A079508
A335109
A354208
A355089
A355262.)

Received May 24 2022; revised versions received January 23 2023; January 24 2023; February
25 2023; April 6 2023.
Published in *Journal of Integer Sequences*,
April 7 2023.

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