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Combinatorial – Voting Counting: How many possible paths exist in which candidate A never has 2 votes ahead?

This question is inspired by Bertrand's ballot theorem. I want to check if I understood the counting method correctly. I changed the initial problem of the ballot theorem a bit.

Suppose we have two candidates, A and B. After counting the votes, we have a tie. How many paths exist in which candidate A is never? $ geq2 $ votes ahead?

This is my approach:

I imagine the election as a path in the $ x $-axis that begins in $ (0, 0) $ and ends in $ (2p, 0) $, where $ 2p $ denotes the total number of votes. In the test of the voting theorem they use a second path that is constructed reflecting in part the original paths on the $ x $-axis. I will try to follow this idea.

First, I change the beginning of all my paths to $ (- 2.2) $ and the first two votes should always be in favor of B. Second, I will extend the way to $ (2p + 2, 2) $ while the last two votes will count for A. Of those paths I will only consider those that begin with two consecutive votes for B and end with two consecutive votes for A (otherwise, we would include paths that are definitely not allowed). Denote $ M $ as the set of those paths. Thus, $ | M | = {2p choose p} $.

Then i build assistant roads as follows:

Allow $ P $ a path of $ M $. While $ P $ does not touch the horizontal line that passes $ (0.2) $ (see red line in the image) Reflect your values ​​on the horizontal line that crosses $ (0.2) $. Those values ​​are the first points of the assistant routes. When $ P $ play the $ (0.2) $-line the assistant paths will follow the rest of $ P $. Building those assistant roads is a bijection in the set of those roads of $ M $ that touch or cross the $ (0, 2) $ line. So I just have to subtract everything assistant paths of $ | M | $.

_ auxiliary route_

Now I count all assistant paths (I will do this with a little more detail):

I have added $ 4 $ votes for $ 2p $ votes from the beginning (see green lines). $ 4 $ votes of each assistant the route always has the same meaning as the first two votes and the last two votes always count for A. Therefore, all assistant the roads add up: $ {2p choose p-2} $. The total number of roads where A is never $ geq 2 $ votes ahead is $ {2p choose p} – {2p choose p-2} = frac {2} {p + 2} {2p choose p} $.

Is this correct?

I appreciate any comments or suggestions and let me know if I should be clearer at some step.

What is the name of the UX pattern to show an ambiguous count of likes, votes, etc.?

While browsing several social media platforms, I find this vote button in favor, which shows an ambiguous count of the things mentioned above.

Is there a name for such a pattern?

enter the description of the image here

[ Politics ] Open question: Is Nancy "the count" Pelosi sure she has the votes for a political trial investigation, since she asked for a vote?

[Politics] Open question: Is Nancy "the count" Pelosi sure she has the votes for a political trial investigation, since she asked for a vote? .

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coinbase transaction: where are the votes of Drivechain calculated?

In the context of the Drivechain proposal, for UTXOs in deposits controlled by miners to be spent, each block must vote on them in a large number of blocks, right?

Now, how are these votes made and where are they calculated? I think the votes are written in some way on the basis of block coins, but how exactly? It is not that there is much space there. And then, each complete node has to keep a table of which transaction has how many votes and verify against each time a transaction is included in the block.

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. – The rich result of AggregateRating does not show the value of the rating, only the votes

I have had the AggregateRating scheme of the last month now and a couple of weeks ago, the stars and votes began to show in the search results. But, one thing is missing and that is the value of the rating.

This is my scheme:

        "@context": "",
        "@type": "TVSeries",
        "aggregateRating": {
            "@type": "AggregateRating",
            "bestRating": "5",
            "worstRating": ".5",
            "ratingCount": "5",
            "reviewCount": "3",
            "ratingValue": "4.9"
        "name": "Name"

This is what I am seeing in the search results.

Ideally, I would like to see the rating precede the votes as "Rating: 4.9 – 5 votes".

What am I doing wrong?

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