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Triangular arrays may list mathematical values other than numbers; for instance the Bell polynomials form a triangular array in which each array entry is a polynomial. [10] Arrays in which the length of each row grows as a linear function of the row number (rather than being equal to the row number) have also been considered. [11]
The remaining positions in each row are filled by a rule very similar to that for Pascal's triangle: they are the sum of the two values to the left and upper left of the position. Thus, after the initial placement of the number 1 in the top row, it is the last position in its row and is copied to the leftmost position in the next row.
These two types of partition are in bijection with each other, by a diagonal reflection of their Young diagrams. Their numbers can be arranged into a triangle, the triangle of partition numbers, in which the th row gives the partition numbers (), (), …, (): [1]
The numbers within the triangle count partitions in which a given element is the largest singleton. The number of partitions of an n-element set into exactly k (non-empty) parts is the Stirling number of the second kind S(n, k). The number of noncrossing partitions of an n-element set is the Catalan number
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This template adds a column of row numbers to a table. The numbers do not sort with the rest of the table, remaining static. Optional classes are provided to display a column label or disable numbers on specific rows.
The Library in 2013. Two further stories of public space and stacks are underground. The Albert and Shirley Small Special Collections Library at the University of Virginia is a research library that specializes in American history and literature, history of Virginia and the southeastern United States, the history of the University of Virginia, Thomas Jefferson, and the history and arts of the ...