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Gordon's Partition Theorem


Let A_(k,i)(n) denote the number of partitions into n parts not congruent to 0, i, or -i (mod 2k+1). Let B_(k,i)(n) denote the number of partitions of n wherein

1. 1 appears as a part at most i-1 times.

2. The total number of appearances of j and j+1 (i.e., any two consecutive integers) together is at most k-1.

Then Gordon's partition theorem states that for 1<=i<=k,

 A_(k,i)(n)=B_(k,i)(n).

The first Rogers-Ramanujan identity corresponds to k=i=2, and the second to k=2, i=1.


See also

Andrews-Gordon Identity, Rogers-Ramanujan Identities

This entry contributed by Andrew Sills

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References

Andrews, G. E. and Santos, J. P. O. "Rogers-Ramanujan Type Identities for Partitions with Attached Odd Parts." Ramanujan J. 1, 91-99, 1997.Gordon, B. "A Combinatorial Generalization of the Rogers-Ramanujan Identities." Amer. J. Math. 83, 393-399, 1961.

Referenced on Wolfram|Alpha

Gordon's Partition Theorem

Cite this as:

Sills, Andrew. "Gordon's Partition Theorem." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/GordonsPartitionTheorem.html

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