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Extending Andrews and Newman's refinement of the crank-mex theorem

George E. Andrews, Brian Hopkins

Abstract

The crank-mex theorem states that the number of integer partitions of $n$ with nonnegative crank equals the number with odd minimal excludant (mex). Andrews and M. Newman recently refined that result in terms of the number of parts greater than one. Here, we establish and expand a complementary result connecting the partitions with even mex, having fixed points, with negative crank, and with positive crank, all refined in terms of number of parts greater than one. We provide both analytic and combinatorial proofs.

Extending Andrews and Newman's refinement of the crank-mex theorem

Abstract

The crank-mex theorem states that the number of integer partitions of with nonnegative crank equals the number with odd minimal excludant (mex). Andrews and M. Newman recently refined that result in terms of the number of parts greater than one. Here, we establish and expand a complementary result connecting the partitions with even mex, having fixed points, with negative crank, and with positive crank, all refined in terms of number of parts greater than one. We provide both analytic and combinatorial proofs.

Paper Structure

This paper contains 7 sections, 3 theorems, 34 equations, 4 tables.

Table of Contents

  1. Introduction
  2. Generating function proof
  3. Combinatorial proof
  4. Bijection between even mex and fixed points
  5. Bijection between fixed points and negative crank
  6. Bijection between negative and positive crank
  7. Remaining open problem

Key Result

Theorem 1

Given positive integers $n$ and $k$, the following are equinumerous: That is, $x_e(n,k) = f(n,k+1) = m_{<0}(n,k) = m_{>0}(n,k+1)$.

Theorems & Definitions (5)

  • Theorem 1
  • Lemma 2
  • proof
  • Lemma 3
  • proof