Solution 73: Schröder Path Pattern Avoidance

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  • Published on Jan 6, 2019
  • We find Schröder paths that avoid UDUDUD, UDUUDD, and HHHHHH by enumerating the desired Dyck paths.
    Congratulations to reynolds45, Zain Majumder, Vampianist3, Nicola C, Rishav Gupta, Kwekinator117, and adandap for successfully solving this math challenge question! reynolds45 was the first person to solve the question.
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Comments • 8

  • Tuanicular
    Tuanicular 5 months ago

    Can you make a solution video to this problem, I’m stumped by it and can’t seem to understand the given solution, so I hope you can show me the process.
    Given f(x) = x^5 - x^3 + 4x, find the integral from 0 to 4 of f^-1(x) wrt x
    Greatly appreciate if you upload a video on this, and always love your vids!

    • dolphin lunggrin
      dolphin lunggrin 3 months ago

      An easy way would be using the fact that the graph of f^-1 is just the graph of f(x) mirrored at the line y=x. or in other words you get the inverse by swapping the x and y axis.
      Since f(0)=0 and f(1)=4, the graph of f is a curve from (0,0) to (1,4). Now sketch the graph between these points and draw a rectangle with the corners (0,0), (0,4), (1,4), (1,0). the upper left part of it is the integral you want and the bottom right part is the integral from 0 to 1 of f(x). The area of the rectangle and the integral of f are easy to calculate and with those you get your solution by taking the difference of the two.
      area of the rectangle is 4
      integral of f is x^6/6 - x^4/4 + 2x^2 and with the bounds 0, 1 you get 23/12
      so the integral from 0 to 4 of f^-1(x) = 4 - 23/12 = 25/12 without ever calculating what the inverse of f even is much less integrating it.

  • Mathedidasko
    Mathedidasko 5 months ago +1

    Hello!
    I am 15 years old and I make math videos, could you please check out my channel? It would be awesome if you subscribed!
    I also have a discord server about math, physics and many other interesting things: discord.gg/njn4hjF
    Anyways, thank you, have a nice day!

  • Nerdiconium
    Nerdiconium 5 months ago +1

    wait, what about one U after the string of D's?

    • LetsSolveMathProblems
      LetsSolveMathProblems  5 months ago +7

      A Dyck path (or a Schröder path) cannot end with a U because its terminal point must be on the x-axis.

  • Kwekinator117
    Kwekinator117 5 months ago +4

    How do you get good at solving these problems?

    • adandap
      adandap 5 months ago +3

      Do lots of them. I worked through all of the previous challenges when I first found this channel (and looked at the answers for those I couldn't get after a good try) and it really helped.

  • Bipin Kumar
    Bipin Kumar 5 months ago

    Thanks sir... it's for jee