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A Young tableau is obtained by filling in the boxes of the Young diagram with symbols taken from some alphabet, which is usually required to be a totally ordered set. Originally that alphabet was a set of indexed variables x 1 , x 2 , x 3 ..., but now one usually uses a set of numbers for brevity.
There are various ways to associate a word (in the sense of combinatorics, i. e., a finite sequence of elements of an alphabet—here the set of positive integers) to every Young tableau. We choose the one apparently most popular: We associate to every Young tableau T the word obtained by concatenating the rows of T from the bottom row to the ...
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For any partition , the Kostka number is equal to 1: the unique way to fill the Young diagram of shape = (, …,) with copies of 1, copies of 2, and so on, so that the resulting tableau is weakly increasing along rows and strictly increasing along columns is if all the 1s are placed in the first row, all the 2s are placed in the second row, and so on.
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Let = be a partition of = + +.It is customary to interpret graphically as a Young diagram, namely a left-justified array of square cells with rows of lengths , …,.A (standard) Young tableau of shape is a filling of the cells of the Young diagram with all the integers {, …,}, with no repetition, such that each row and each column form increasing sequences.
This forms 26 rows of shifted alphabets, ending with an alphabet starting with Z (as shown in image). Separate from these 26 alphabets are a header row at the top and a header column on the left, each containing the letters of the alphabet in A-Z order. The tabula recta can be used in several equivalent ways to encrypt and decrypt text.
The I.T.A. originally had 43 symbols, which was expanded to 44, then 45. Each symbol predominantly represented a single English sound (including affricates and diphthongs), but there were complications due to the desire to avoid making the I.T.A. needlessly different from standard English spelling (which would make the transition from the I.T.A. to standard spelling more difficult), and in ...
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