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Ferrers Diagram


FerrersDiagram

A Ferrers diagram represents partitions as patterns of dots, with the nth row having the same number of dots as the nth term in the partition. The spelling "Ferrars" (Skiena 1990, pp. 53 and 78) is sometimes also used, and the diagram is sometimes called a graphical representation or Ferrers graph (Andrews 1998, p. 6). A Ferrers diagram of the partition

 n=a+b+...+c,

for a list a, b, ..., c of k positive integers with a>=b>=...>=c is therefore the arrangement of n dots or square boxes in k rows, such that the dots or boxes are left-justified, the first row is of length a, the second row is of length b, and so on, with the kth row of length c. The above diagram corresponds to one of the possible partitions of 100.

The Ferrers diagram of a given partition n is implemented in the Wolfram Function Repository as ResourceFunction["FerrersDiagram"][n].

YoungDiagramLatticePaths

The partitions of integers less than or equal to mn in which there are at most n parts and in which no part is larger than m correspond (1) to Young tableaux which fit inside an m×n rectangle and (2) to lattice paths which travel from the upper right corner of the rectangle to the lower left in m+n leftward and downward steps. The number of Young diagrams fitting inside an m×n rectangle is given by the binomial coefficient (m+n; m)=(m+n; n). The above example shows the

 (2+2; 2)=(4; 2)=(4!)/(2!2!)=(24)/4=6

Young 2×2 diagrams.


See also

Conjugate Partition, Durfee Square, Self-Conjugate Partition, Staircase Walk

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References

Andrews, G. E. The Theory of Partitions. Cambridge, England: Cambridge University Press, pp. 6-7, 1998.Comtet, L. "Ferrers Diagrams." §2.4 in Advanced Combinatorics: The Art of Finite and Infinite Expansions, rev. enl. ed. Dordrecht, Netherlands: Reidel, pp. 98-102, 1974.Liu, C. L. Introduction to Combinatorial Mathematics. New York: McGraw-Hill, 1968.MacMahon, P. A. Combinatory Analysis, Vol. 2. New York: Chelsea, pp. 3-4, 1960.Propp, J. "Some Variants of Ferrers Diagrams." J. Combin. Th. A 52, 98-128, 1989.Riordan, J. An Introduction to Combinatorial Analysis. New York: Wiley, pp. 108-109, 1980.Skiena, S. "Ferrers Diagrams." §2.1.2 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 53-55, 1990.Stanley, R. P. Enumerative Combinatorics, Vol. 1. Cambridge, England: Cambridge University Press, 1999.Stanton, D. and White, D. Constructive Combinatorics. New York: Springer-Verlag, 1986.

Referenced on Wolfram|Alpha

Ferrers Diagram

Cite this as:

Weisstein, Eric W. "Ferrers Diagram." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FerrersDiagram.html

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