🔔 Bell Numbers and Its Recurrence Relation (Proof)

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  • Published on Sep 23, 2018
  • We define Bell Numbers and derive its recurrence relation.
    For exponential generating function: usclip.net/video/1I4VpemnWmg/video.html
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Comments • 14

  • Sumit Kumar
    Sumit Kumar 4 months ago

    Excellent.

  • Duvvada Jayanth Kumar
    Duvvada Jayanth Kumar 8 months ago +1

    Tq very much

  • James Wilson
    James Wilson 9 months ago +1

    Sweet! I love this kind of stuff!

  • Vighnesh Raut
    Vighnesh Raut 9 months ago +1

    When you said partitions, I was thinking about the partitions that Ramanujan generalised. But these are different.

  • Konrad Kamiński
    Konrad Kamiński 9 months ago +3

    According to given definition B(0) =0 there is no partitioning of empty set without using empty set, but theorem works only when we set B(0)=1.

    • Konrad Kamiński
      Konrad Kamiński 9 months ago

      Or you can say that partition is maximal set of disjoint subsets. It doesn't change B for n greater than 0 but it defines B(0)=1 correctly. But then we have to live with every partition containing empty set. ;) Ehh, life is hard!

    • LetsSolveMathProblems
      LetsSolveMathProblems  9 months ago +3

      One may argue that we can consider {empty set} as a "partition" of empty set, but you are right--from the definition I gave, B(0) = 0. Perhaps I should have mentioned that our definition applies only for positive integer n, then defined B(0) as 1 at the beginning of the video.

  • Logan Hargrove
    Logan Hargrove 9 months ago +3

    Thank you! I have been working on a certain infinite series and I just now realized that the result can be expressed in terms of Bell numbers.

  • Paco Gomez-Paz
    Paco Gomez-Paz 9 months ago

    I am trying to prove that the cardinality of a bell set is greater than that of the original set for both infinite and non infinite sets non infinite is quite easy to prove but I am having troubles for infinite sets . Anyone have any ideas

    • Paco Gomez-Paz
      Paco Gomez-Paz 8 months ago

      I meant greater than or equal to also I already proved it but thank you for taking the time to respond

    • Konrad Kamiński
      Konrad Kamiński 9 months ago

      What is Bell set? Is it set of all partitions? Then it's not true for n=1? For infinite just notice that Bell set's power is greater or equal than Power set's cardinality (using partitions of 2 sets) and Power set's cardinality is strictly greater than cardinality of original set.
      If Bell's set is something different, plz tell your definition.
      PS. Actually now I see that B(Kappa)=2^(Kappa) for Kappa greater or equal Aleph_0.

  • adandap
    adandap 9 months ago +6

    Nice, though essentially an induction proof.

  • Xander Gouws
    Xander Gouws 9 months ago +11

    Your vids are really warming me up to combinatorics

  • GENIUS
    GENIUS 9 months ago

    Awesome!!!!!!I absolutely love your videos!!!Keep them coming.Oh,and by the way,could you solve this integral that's been bugging me:int from 0 to pi/4 of(1-(x)^2+(x)^4-(x)^6...)/((cosx)^2+(cosx)^4+(cosx)^6+....).You can find it here around the 1:34:00 mark in the integration bee here: usclip.net/video/UNpa3-EGxGY/video.html