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combinat[conjpart] - conjugate partition

Calling Sequence

conjpart(p)

Parameters

p

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partition; non-decreasing list of positive integers

Description

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The conjpart(p) command computes and returns the conjugate partition of p.

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A partition p=[i1,i2,...,im] of a positive integer n may be represented visually by its Ferrer's diagram. This is a diagram composed of dots in rows, in which the kth row consists of ik dots, for k=1..m. The total number of dots in the diagram is equal to the number n. For example, the partition 2,3,5 of 10 has the Ferrer's diagram:

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consisting of ten dots arranged in three rows, with two dots in the first row, three dots in the second, and five dots in the third row.

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Two partitions (of a positive integer n) are said to be conjugates if their Ferrer's diagrams are conjugate, which means that one is obtained from the other, by reflection along the anti-diagonal, by writing the rows as columns and columns as rows. For example, the conjugate of the Ferror diagram above is:

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which represents the partition 1,1,2,3,3. Therefore, the partitions 2,3,5 and 1,1,2,3,3 are conjugate partitions.

Examples

withcombinat:

conjpart2,3,5

1,1,2,3,3

(1)

conjpart1,1,2,3,3

2,3,5

(2)

conjpart1,2,3

1,2,3

(3)

See Also

combinat[encodepart], combinat[inttovec], combinat[numbpart], combinat[partition], combinat[randpart], Definition/partition


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