Geodesic
Lattice points and simultaneous core partitions
Paul Johnson1
1University of Sheffield
The electronic journal of combinatorics, Tome 25 (2018) no. 3
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We apply lattice point techniques to the study of simultaneous core partitions. Our central observation is that for $a$ and $b$ relatively prime, the abacus construction identifies the set of simultaneous $(a,b)$-core partitions with lattice points in a rational simplex. We apply this result in two main ways: using Ehrhart theory, we reprove Anderson's theorem that there are $(a+b-1)!/a!b!$ simultaneous $(a,b)$-cores; and using Euler-Maclaurin theory we prove Armstrong's conjecture that the average size of an $(a,b)$-core is $(a+b+1)(a-1)(b-1)/24$. Our methods also give new derivations of analogous formulas for the number and average size of self-conjugate $(a,b)$-cores.
DOI : 10.37236/5734
Classification : 11P81, 52B20, 05A17
Mots-clés : partitions, rational Catalan combinatorics, Ehrhart theory

Paul Johnson  1

1 University of Sheffield 
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Paul Johnson. Lattice points and simultaneous core partitions. The electronic journal of combinatorics, Tome 25 (2018) no. 3. doi: 10.37236/5734

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