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Knowledge Zone<a href="https://mstdn.social/tags/DidYouKnow" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#DidYouKnow</a>: In mathematics, <a href="https://mstdn.social/tags/PascalsTriangle" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#PascalsTriangle</a> is a triangular array of the binomial coefficients that arises in probability theory, combinatorics, and algebra.Pascal's triangle determines the coefficients which arise in binomial expansions<a href="https://knowledgezone.co.in/kbits/6429997fc2045832df405e50" rel="nofollow noopener noreferrer" translate="no" target="_blank">https://knowledgezone.co.in/kbits/6429997fc2045832df405e50</a>

bs2> We all know and love <a href="https://mstdn.jp/tags/PascalsTriangle" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#PascalsTriangle</a> and the <a href="https://mstdn.jp/tags/FibonacciSequence" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#FibonacciSequence</a>. But who knows <a href="https://mstdn.jp/tags/meruprasthara" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#meruprasthara</a> and the Pingala sequence?.. America invented <a href="https://mstdn.jp/tags/ModernDemocracy" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#ModernDemocracy</a>!.. Even though the <a href="https://mstdn.jp/tags/Haudenosaunee" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#Haudenosaunee</a> practiced it for centuries before on the same land. Let us be clear: we are very bad at attribution. Citing our sources is an echochamber, a way to further empower voices that are already empowered, whether they did the work and contributed the value or not. <a href="https://drym.org/on-attribution/" rel="nofollow noopener noreferrer" target="_blank">https://drym.org/on-attribution/</a> <a href="https://mstdn.jp/tags/PingalaSequence" class="mention hashtag" rel="nofollow noopener noreferrer" target="_blank">#PingalaSequence</a>

PensionDanA more elegant path forward dawned on me (using <a href="https://mathstodon.xyz/tags/PascalsTriangle" class="mention hashtag" rel="tag">#PascalsTriangle</a>!), so I won't be using the equation in the preceding post. First, some comments & definitions. Cycles will refer to cycles of length 2 or more (not including 1-cycles, which we will regard as positions along the main diagonal that don't get permuted). Define the delta of two consecutive elements of a cycle a, b as b - a. Clearly, the sum of all the deltas of a cycle is zero, as each element is added and subtracted once. Therefore, every cycle has at least one negative delta among its pairs of consecutive elements. Define an increasing cycle as a cycle with just one negative delta. Observe that an increasing cycle can be written with its elements in increasing order (the delta from the last element back to the first element is the negative one). Consider these sets of positions NEZ and SWZ (see illustration) in an n x n matrix, which lie in the antidiagonal and the sub-antidiagonal (thus in the eligible positions for (\Omega^{xx}_{n}\)): • NEZ; the northeast zigzag of (n-1) positions that begins at the upper right corner, ordered as positions \(a_{1,n}, a_{2,n}, a_{2,n-1}, a_{3,n-1},…a_{[(n+1)/2],[(n+1)/2]+1}\). • SWZ; the southwest zigzag of (n-1) positions that begins at the lower left corner, ordered as positions \(a_{n,1}, a_{n,2}, a_{n-1,2}, a_{n-1,3},…a_{[(n+1)/2]+1,[(n+1)/2]}\). We’ll describe a set of cycles C, and then show that permutations which are a product of disjoint cycles from C are vertices of \(\Omega^{xx}_{n}\). As we know that \(\Omega^{xx}_{n}\) has \(2^{n-1}\) vertices, if we can verify that there are \(2^{n-1}\) such permutations, then we will have characterized all the vertices of \(\Omega^{xx}_{n}\).

R. Sunder-RajSome stuff that led to me thinking about a relationship between <a href="https://mathstodon.xyz/tags/PascalsTriangle" class="mention hashtag" rel="tag">#PascalsTriangle</a> and <a href="https://mathstodon.xyz/tags/binaryTrees" class="mention hashtag" rel="tag">#binaryTrees</a>. It seems to have started with me trying to represent Pascal’s triangle using things like <a href="https://mathstodon.xyz/tags/crochet" class="mention hashtag" rel="tag">#crochet</a>, <a href="https://mathstodon.xyz/tags/macrame" class="mention hashtag" rel="tag">#macrame</a>, <a href="https://mathstodon.xyz/tags/braiding" class="mention hashtag" rel="tag">#braiding</a> and <a href="https://mathstodon.xyz/tags/knitting" class="mention hashtag" rel="tag">#knitting</a>. <a href="https://mathstodon.xyz/tags/mathart" class="mention hashtag" rel="tag">#mathart</a> <a href="https://mathstodon.xyz/tags/mathsart" class="mention hashtag" rel="tag">#mathsart</a> <a href="https://mathstodon.xyz/tags/binaryTree" class="mention hashtag" rel="tag">#binaryTree</a>

R. Sunder-RajAnd another clip of a representation of a <a href="https://mathstodon.xyz/tags/PascalsTriangle" class="mention hashtag" rel="tag">#PascalsTriangle</a> <a href="https://mathstodon.xyz/tags/BinaryTree" class="mention hashtag" rel="tag">#BinaryTree</a><a href="https://mathstodon.xyz/tags/mathart" class="mention hashtag" rel="tag">#mathart</a>

R. Sunder-RajHere is a clip to do with the <a href="https://mathstodon.xyz/tags/PascalsTriangle" class="mention hashtag" rel="tag">#PascalsTriangle</a> <a href="https://mathstodon.xyz/tags/BinaryTree" class="mention hashtag" rel="tag">#BinaryTree</a> shown as individual paths layered together<a href="https://mathstodon.xyz/tags/mathart" class="mention hashtag" rel="tag">#mathart</a>

R. Sunder-RajI compiled these ideas into a Twitter moment<a href="https://mathstodon.xyz/tags/PascalsTriangle" class="mention hashtag" rel="tag">#PascalsTriangle</a><a href="https://twitter.com/i/moments/992234400715862016" target="_blank" rel="nofollow noopener noreferrer" translate="no">https://twitter.com/i/moments/992234400715862016</a>

R. Sunder-RajSplitting a <a href="https://mathstodon.xyz/tags/PascalsTriangle" class="mention hashtag" rel="tag">#PascalsTriangle</a> <a href="https://mathstodon.xyz/tags/BinaryTree" class="mention hashtag" rel="tag">#BinaryTree</a> into two copies.<a href="https://mathstodon.xyz/tags/mathart" class="mention hashtag" rel="tag">#mathart</a>

R. Sunder-RajThese ideas eventually worked their way into a <a href="https://mathstodon.xyz/tags/woven" class="mention hashtag" rel="tag">#woven</a> <a href="https://mathstodon.xyz/tags/BinaryTree" class="mention hashtag" rel="tag">#BinaryTree</a> representation of <a href="https://mathstodon.xyz/tags/PascalsTriangle" class="mention hashtag" rel="tag">#PascalsTriangle</a>.<a href="https://mathstodon.xyz/tags/mathart" class="mention hashtag" rel="tag">#mathart</a>

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