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For the original, see Traité du triangle arithmétique, 1654; thank you Cambridge University.
Blaise Pascal discussed combinatorial numbers of n things taken m at a time and determination of the probability of winning games by the numbers of his triangle in his Traité du triangle arithmétique, 1654. The numbers were also known as terms for binomial expansions of powers. CLICK for graphics on the connection of Pascal's Triangle to Binomial Theorem (or the descriptions of values of a power by formulas depending on rows of the triangle).
Cool truism easily visible in Pascal's Triangle:
The total sum of rows' values (which each equal 2^n) equals 2^(n+1) - 1. Look at a sum of the 1st through xth of a diagonal series as it appears as the xth of the next diagonal. Add this type of summation across an entire row-level of the triangle excepting one "1" (because it does not represent the sum of any diagonal) and get the cumulative value of the triangle up to that row. DEFINITIONS OF THE FIGURATE NUMBERS:Polytopic Figurate Numbers (of "increasing dimension") fill Pascal's Triangle. They are the series that are the diagonals from the top edges of Pascal's Triangle. These are distinguishable from other series that are also figurate called polygonal numbers (which are fundamentally two-dimensional and compositions of triangular numbers). See Math World figurate number and polygonal number.
Values diagrammed:
Recurrence/shell definition of PT figurate numbers: xth n-PT diagonal number = each of 1 through x summed of the (n-1)-PT diagonal numbers. Here is one visual explanation of the triangle (and my illegitimate name): N! divided by !И for only the "first" m/n number of the factorial sets' multiples
The usual definition of the mth of row n is: C(n,m) = n! / (n - m)! m! where n is the row (don't count the top single-box) and m is the count ("1" starts with the second box in.) Here are ways that eye like to write C(n,m) since they are more graphically related to the above.
Here are comprehensive formulas for any xth n-PT diagonal number:
Notice that, for as much as a nth PT figurate series is determined, likewise, the xth number of that many PT figurate series is determined. (The xth of n-PT diagonal equals the nth of x-PT diagonal.)
A result of the triangle's obvious symmetry is, any two neighboring figurate number series certainly share one number in common: the (n+1)th of the n-PT diagonal and the nth of the (n+1) diagonal.
Here is a chart of proportional relationships as occur between figurate numbers (first defined by Gerolamo Cardano in 1570, Edwards p. 43.)
The PT diagonals might also be called simplices. That name is from geometry. The shape that is a n-simplex shape is some level of: line, triangle, tetrahedron, pentatope, or higher level of hypertetrahedron. This entire class of hyper-geometric n-simplex figures always exhibits a certain number of each and every lower-level-than-itself tetrahedral-generalization within its own structure. The number of generalized tetrahedral figures per level of n-simplex dimension, taken by ordered size, single classes of structures, will generate the Pascal Triangle figurate series, class by class. This is explained much more clearly by Eric Rowland in his web page "Polytopes". Polytopes become difficult to envision. In comparison, polygonal figurate numbers chart easily and 2-dimensionally. This is because polygonal figures are simply sets of triangular numbers. In comparison, levels of polytope number represent sets of sets of sets of... of triangular number. The diagonals of Pascal's Triangle are full of "increase in dimension". Pascal's Triangle might be describing possible coin toss outcomes or sets of True/False answers.
Pascal's triangle may describe the accumulation of directions such as used in vectors.
Pascal's Triangle might be describing types of set--size-groupings of things, when one new thing is added per row like this:
Pascal's Triangle is about internal definitions of a value, row by row adding one "variable that is one", thereby creating expanding varieties of definition by including the new variable of "one" in combination with previous "ones". The Pascal Triangle considers sets that are combinations of x things, without having the order of things matter. Here is a visual representation of Pascal's Triangle values with letter-set content per unit counted: Pink = a; red = b; orange = c; gray = d; yellow = e; green = f. Pascal Triangle values are the value of the count of solid colored, black edged squares. A letter-set that represents the unit to be counted, e.g., bd or cdf or abcde, is visually decipherable in the below chart because a letter is assigned for each layer of color underneath a unit (in addition to the unit's color). Notice that backgrounds are not counted since they are only descriptive of a square that is counted.
Notice that the cumulative to row n, sum total-amount of order-undifferentiated groups of n things, is: the sum of 0 through n values when applied to the equation 2^n. This depiction//white = a; red = b; orange = c; purple = d; yellow = e; green = f; aqua = g. Notice that each letter lays out its own Pascal's Triangle within the triangle.
Notice that the xth value of the (n+1)th figurate series is either: the sum of the 1st through xth of size-n sets, OR, the sum of the 1st through nth of size-(x-1) sets. "k-subset"If "sets of size-k content" (or a k-subset in combinatorics) will be examined, the amount of sets that exist of size-k depends on how many elements exist to pull k out of. The total amount of "the-sets-of-size-k", given n elements with which to form sets, increases at the (k+1)th Pascal Triangle-figurate rate of growth. A total amount of k-sets is summed at the (k+1)-PT-figurate rate. Pascal Triangle figurate series and values of powers:The above explains why the power of n always relates to the (n+1)-Pascal Triangle-figurate series as in Worpitzky's identity of 1883 (see Euler/Pascal definitions). The xth of (n+1)-PT that is important for and tied to definitions of power values is apparently the summed count of groups of size-n by the (x+n-1)th level of variables included within sets. Euler Triangle's row n value is definable as "all possible" orderliness as regards permutations of n things. So the power of n is Euler's row n multiplying applicable sets of size-n, thus counting up all of the permutations of certain size-n groups. For x^n, each of the values of Euler row n multiplies the value that is, firstly, the (x+n-1)-elements/source level of size-n-groups, then each of the next (n-1) lower levels of source elements creating size-n groups. Euler Triangle values and values of powers:Euler Triangle row n concerns all ordered permutations of n things. By addressing as multiples, n levels of all possible sized-n groups taken firstly out of (x+n-1) amount of elements; then as taken from (x+n-2) elements; then taken from less one again; etc., continuing to reduce the total elements from which sized-n groups are drawn by one per category of orderliness of n ordered things (in other words, per value seen in a Euler Triangle row): an x^n value is defined. |
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