Number of arrangements of of $1$ and $-1$ such that sum is always non-negative at any point of addition and overall sum is $0$.

Asked Jul 27 '20 at 08:49

Active Jul 27 '20 at 19:50

Viewed 125 times

Can you think of the number of ways of adding "$1$" and "$-1$" (where both are used equal number of times) such that the at any point during the addition the sum is non-negative and the overall sum is 0. For e.g.

$a. 1 - 1 + 1 - 1 = 0 \rightarrow$ correct

$b. 1 - 1 - 1 + 1 = 0 \rightarrow$ incorrect (As sum becomes $-1$ i.e. negative at third element)

$c. 1 + 1 + 1 - 1 - 1 + 1 - 1 - 1 = 0 \rightarrow$ correct

$d. 1 + 1 - 1 - 1 - 1 + 1 = 0 \rightarrow$ incorrect (As sum becomes $-1$ i.e. negative at fifth element)

edited Jul 27 '20 at 19:50

RobPratt

50,938

asked Jul 27 '20 at 08:49

Parth Wadhawan

3

https://math.stackexchange.com/questions/2991347/catalan-numbers-sequence-of-balanced-parentheses – YJT Jul 27 '20 at 08:53
2

See also this wiki page, or this one for more on your specific problem. Note that proofs $1,3,5,6$ on the first wiki page use the definition of the Catalan numbers that corresponds to this problem. – Ben Grossmann Jul 27 '20 at 09:00
Correction: proof 2 also uses an analogous defintion – Ben Grossmann Jul 27 '20 at 09:06

Number of arrangements of of $1$ and $-1$ such that sum is always non-negative at any point of addition and overall sum is $0$.

0 Answers0