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Binomial Theorem

Basis for dualities?

• x and y (division)
• i and j
• xy and yx
• Wikipedia: Pascal's triangle
• Binomial theorem
• Binomial coefficient
• Multinomial theorem
• List of factorial and binomial topics

How does the binomial theorem relate to the triangle of power? (The relationship between a, b, c in {$a=b^c$}?)

Examples

• 2 x 2 square - conceptual factors - "and" and "or"
• eightsome - logical square 2 x 2 vertices and 4 pairs (edges)
• 2 x 2 multiplication
• 3 x 3 factorial experiment - Norman Anderson - multinomial
• girl + boy
• genetics - dominant and recessive
• truth tables
• expand 1.9 to a power = 2 - .01
• |A union B| + |A intersect B| = |A| + |B|
• Russian roulette
• interest rates - and taking the limit - formula for {$e^x$}
• simplexes, cross polytopes, hypercubes, coordinate systems

Infinite Series. Dissecting Hypercubes with Pascal's Triangle.

3 Blue 1 Brown. Derivative formulas through geometry.

Product rule {$u\cdot v + \textrm{d}(u\cdot v) = (u + \textrm{d}u)(v + \textrm{d}v)$}

• General Leibniz rule

Proof of the binomial theorem

• By bijection
• By induction
• By calculus
  • Eddie Woo. An ingenious & unexpected proof of the Binomial Theorem (1 of 2: Prologue)
  • Eddie Woo. An ingenious & unexpected proof of the Binomial Theorem (2 of 2: Proof) Considering the differentiation of {$(x+1)^n$} in two different ways, before expanding and after expanding.

Sum of the row is {$2^n$}.

Inclusion-exclusion principle Counting nondivisible integers. Counting derangements. Number of surjective functions. Stirling numbers of the second kind. Rook polynomials. Main proof and algebraic proof. Combinatorial sieve methods.(Brun.)

• Boole's inequality Bonferroni inequalities
• Maximum-minimums identity
• Schuette-Nesbitt formula
• De Moivre's formula {$(\textrm{cos} \;x + i \;\textrm{sin}\; x)^n = \textrm{cos}\; nx + i \; \textrm{sin}\; nx$} can be used to calculate roots, or to get formulas for multiples of angles.
  • Gives Chebyshev polynomials.
• Catalan number {$(n+1)C_n=\binom{2n}{n}$}
  • Extremal theory of set systems. Sperner showed that the maximum possible size of a collection of subsets of an n-element set such that no set is a subset of another is the largest binomial coefficient, {$\binom{n}{\frac{n}{2}}$} if n is even and {$\binom{n}{\frac{n+1}{2}}$} if n is odd.
• Pushdown automaton
• Mandelbrot set
• Hypergeometric function
• Binomial series
  • Black pen red pen. Can we really do "-2 choose 5"? Generalized Binomial Theorem.
• Fermat's little theorem
  • {$(x+a)^n\equiv x^n+a^n \equiv x^n + a (\textrm{mod}\; n)$} where n is prime, a is any integer, tests for primality in polynomial time
  • Euler's theorem
  • Carmichael function
  • Lagrange's theorem in group theory
• Elementary symmetric functions
• Basis for a Clifford algebra
• Exterior algebra, exterior product, {$e_i\wedge e_i=0$}, determinant, area in the plane
• Umbral calculus
• Euler's characteristic {$-1 + V -E + F -1 = 0$}
• Betti numbers
• Gaussian binomial coefficient
• Binomial distribution
  • binomial random variable
  • Central limit theorem
• Poisson distribution
• Negative binomial distribution (Meixner polynomials) A discrete probability distribution that models the number of failures in a sequence of independent and identically distributed Bernoulli trials before a specified (non-random) number of successes (denoted r r) occurs. (Russian roulette.) {$P\left(\bigcap_{t\in S} X_t^C\right) = (1-p)^{|S|} = \sum_{n=0}^{|S|} {|S| \choose n} (-p)^n$}. An upper bound for this quantity is {$e^{-p|S|}$}
• Normal distribution

The coefficient of {$q^r$} in {$\binom{n}{k}_q$} counts the number of inversions, for example, transforming 00011 to 01100 so that 1 is to the left of 0, here in 4 ways.

The coefficient of {$q^r$} in {$\binom{n+m}{m}_q$} counts the number of partitions of r with m or fewer parts each less than or equal to n.

{$\binom{n}{k}_q$} counts the number of k-dimensional vector subspaces of an n-dimensional vector space over {$F_q$} (a Grassmannian).

When k=1 and q=1 we have the Euler characteristic of the complex Grassmannian, and when q=-1, the Euler characteristic of the real Grassmannian.

The number of k-dimensional affine subspaces of {$F_{q^n}$} is {$q^{n-k}\binom{n}{k}_q$}. This allows an interpretation of the identity {$\binom{m}{r}_q=\binom{m-1}{r}_q + q^{m-r}\binom{m-1}{r-1}_q$}.

• Grassmannian Key to Moore's proof of Bott periodicity.
• God's Binomial Identity Expressing a number {$d=d_1*d_2*d_3$} as the sum of signed binomial coefficients {$d=\binom{d_1+d_2+d_3}{3}-\sum_{i<j}\binom{d_i+d_j}{3}+\sum_{i}\binom{d_i}{3}-\binom{0}{3}$}

Directed simplicial complexes

• Infinite Series: Your Brain As Math: Part 1
• Infinite Series: Your Brain As Math: Part 2: Simplicial Complexes and Betti Numbers
• Infinite Series: Your Brain As Math: Part 3: Your Mind is Eight-Dimensional

Finite field with one element

Grassmannians

• Michael Attiyah: Classical projective geometry: the twisted cubic, the quadric surface, or the Klein representation of lines in 3-space. These illustrate, respectively, the theory of rational curves, of homogeneous spaces, and of Grassmannians. The classical conic is a rational curve, a quadric and a Grassmannian all in one.
  • Any conic lies on some Gr(2,4).
• Richard Borcherds. Algebraic Geometry 20 Grassmannians.
• Dev Sinha. Algebraic Topology From Geometric Perspective
  • Dev Sinha. Lecture 5: Characteristic Classes Develops characteristic classes through Thom cochains. Develops the homology and cohomology of infinite Grassmannians. Touches on Bott periodicity.
  • Lecture 7: Iterated Loop Spaces
  • Lecture 9: Homology and Cohomology of Symmetric Groups, Part 1
• Andrei Okounkov. Enumerative geometry and geometric representation theory. (Stable envelopes and K-theory.)
  • Lecture 7 Mentions Bott periodicity.
• Positive Grassmanian is the subset of the real Grassmannian where all Plücker coordinates are nonnegative
  • Amplituhedron By calculating its volume, in effect you’re calculating the amplitude for the given collision.
    • Arkani-Hamed and his collaborators had defined the amplituhedron in relation to the positive Grassmannian. They demonstrated that it’s possible to change a positive Grassmannian into the amplituhedron by multiplying it by a type of matrix, effectively providing a mathematical recipe for moving points on the positive Grassmannian over to points on the amplituhedron. As a result, information about the relatively well-studied positive Grassmannian transfers to the relatively unexplored amplituhedron.
    • Nima Arkani-Hamed and Thomas Lam in conversation
  • A Mathematician’s Unanticipated Journey Through the Physical World
  • Lauren K. Williams. The positive Grassmannian, the amplituhedron, and cluster algebras
  • Williams wrote a series of papers with the mathematician Sylvie Corteel that explored an unexpected link between the combinatorics of the positive Grassmannian and statistical physics.
  • Solitons and the Kadomtsev-Petviashvili equation. {$\partial_x(\partial_t u+u \partial_x u+\epsilon^2\partial_{xxx}u)+\lambda\partial_{yy}u=0$} where {$\lambda=\pm 1$}. A one-to-one relationship between points on the positive Grassmannian and solutions to the KP equation. The large-scale behavior of a wave formation is entirely determined by which cell your point in the positive Grassmannian lies in.
  • Planar bicolored graphs.

Bottom-up and top-down duality

• Root systems
• Poincare duality
• Jordan curve theorem
• Riemann-Roch theorem
• Atiyah–Singer index theorem
• Hodge star operator ?