A bijective proof of a theorem on 3-core partitions (Q2816449)

A 3-core partition is a partition such that none of the hook lengths are divisible by \(t\). Let \(a_3(n)\) denote the number of 3-core partitions of \(n\). Given a partition \(n=x_1+\cdots+x_k\) with \(x_j\geq x_{j+1}\) for \(j=1,\ldots,n-1\), the author introduces the ``word'' associated to the partition as follows. Starting at the bottom row of the Ferrers diagram, the number for each row is the difference between the number of dots in the row and the number of dots in the row below it. A word is a 3-core word if its associated partition is a 3-core partition. The aim is to establish a bijection between the 3-core partitions and the 3-core words and to provide bijective proofs of some known identities for 3-core partitions.NEWLINENEWLINEThe first observation is that every 3-core word has the form \(W=A_rB_s\) where \(A_r\) is the \(r\) letter subword \(101010\ldots\) and \(B_s\) is the \(s\) letter subword \(222\ldots\), and conversely. Since the partition has no hooks of length 3, a few diagrams show that the associated words only use the symbols 0, 1, 2, the symbol 2 must be followed by 2 and the sequences 00 and 11 cannot appear. This characterisation is used to give a bijective proof of a result of \textit{M. D. Hirschhorn} and \textit{J. A. Sellers} [Bull. Aust. Math. Soc. 79, No. 3, 507--512 (2009; Zbl 1183.05009)]. Namely, if \(p\) is a prime congruent to 2 modulo 3 and \(k\) is even, then \(a_3(n) = a_3(p^kn + (p^k-1)/3)\). It is enough to prove this for \(k=2\) because the general case follows by induction. The basis of the construction is the function \(f\) which takes a 3-core word \(W=A_rB_s\) to another 3-core word: NEWLINE\[NEWLINEf(W)=f(A_rB_s)=\begin{cases} A_{pr+(p-2)/3}B_{ps+(2p-1)/3}&\text{ if }r \text{ is odd}\\ A_{pr+(2p-1)/3}B_{ps+(p-2)/3}&\text{ if }r\text{ is even}.\end{cases}NEWLINE\]NEWLINE If \(W=A_rB_s\) represents a partition of \(n\), then \(f(W)\) represents a partition of \(p^2n+(p^2-1)/3\). Another observation is that the number of dots \(D(r,s)\) in the Ferrers diagram for the partition corresponding to \(W=A_rB_s\) is given by a pair of quadratic forms in \(r\) and \(s\). Some ingenious manipulations lead to the identity \(a_3(n) = a_3(p^2n+(p^2-1)/3)\). The author also derives another result of Hirschhorn and Sellers [loc. cit.] by evaluating \(a_3(n)\) in terms of the prime factorisation of \(3n+1\), This is done by counting the number of solutions for the binary quadratic forms arising from \(3n+1=3D(r,s)+1\) to find the prime factorisation of \(3n+1\) in \(\mathbb{Z}[(1+\sqrt{-3})/2]\).NEWLINENEWLINEFinally, the author speculates on an extension of the approach to \(t\)-core words. It appears that \(t\)-core words have \((n-2)!\) allowable combinations of substrings (rather than just the single combination \(A_rB_s\) in the case \(t=3\)) and it seems much more difficult to describe the structure of \(t\)-core words.