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Triangular array!

Source: Romania TST 2013 Day 5 Problem 3

January 21, 2015

combinatorics unsolvedcombinatorics

Problem Statement

Given a positive integer

n

, consider a triangular array with entries

a_{ij}

where

i

ranges from

1

to

n

and

j

ranges from

1

to

n-i+1

. The entries of the array are all either

a_{i-1,j} = a_{i-1,j+1}

or

1

, and, for all

i > 1

and any associated

j

,

a_{ij}

is

0

if

a_{i-1,j} = a_{i-1,j+1}

, and

a_{ij}

is

1

otherwise. Let

S

denote the set of binary sequences of length

n

, and define a map

f \colon S \to S

via

f \colon (a_{11}, a_{12},\cdots ,a_{1n}) \to (a_{n1}, a_{n-1,2}, \cdots , a_{1n})

. Determine the number of fixed points of

f

.

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