\((-1)\)-enumeration of self-complementary plane partitions (Q1773197)

A plane partition \(P\) can be defined as a finite set of points \((i,j,k)\) with \(i,j,k>0\) and if \((i,j,k) \in P\) and \(1\leq i'\leq i\), \(1\leq j'\leq j\), and \(1\leq k'\leq k\), then \((i',j',k') \in P\). If \(i\leq a\), \(j\leq b\), and \(k\leq c\), then the plane partition \(P\) is said to be contained in the box with sidelengths \(a,b,c\). MacMahon enumerated the number of plane partitions contained in the box with sidelengths \(a,b,c\) as \[ B(a,b,c)=\prod_{i=1}^a {(c+i)_b\over (i)_b}, \] where \((x)_n=x(x+1)\cdots (x+n-1)\), the rising factorial function. A plane partition \(P\) is called self-complemantary, if \((i,j,k)\in P \leftrightarrow (a+1-i,b+1-j,c+1-k)\notin P\) for \(1\leq i\leq a\), \(1\leq j\leq b\), \(1\leq k\leq c\). Stanley showed that the number SC\((a,b,c)\) of self-complementary plane partitions contained in the box with sidelengths \(a,b,c\), is equal to \(B({a\over 2}, {b\over 2}, {c\over 2})^2\) for \(a,b,c\) even; \(B({a\over 2}, {b+1\over 2}, {c-1\over 2})B({a\over 2}, {b-1\over 2}, {c+1\over 2})\) for \(a\) even and \(b,c\) odd; \(B({a+1\over 2}, {b\over 2}, {c\over 2})B({a-1\over 2}, {b\over 2} , {c\over 2})\) for \(a\) odd and \(b,c\) even. (Clearly there is no self-complementary plane partition contained in the box with sidelengths \(a,b,c\), if all \(a,b,c\) are odd.) The current paper enumerates self-complementary plane partitions contained in the box with sidelengths \(a,b,c\), with weights \((-1)^{n(P)}\). Up to sign, the result is \(B({a\over 2}, {b\over 2}, {c\over 2})\) for \(a,b,c\) even; \(\text{SC} ({a\over 2}, {b+1\over 2}, {c-1\over 2}) \text{SC}({a\over 2}, {b-1\over 2}, {c+1\over 2})\) for \(a\) even and \(b,c\) odd; \(\text{SC}({a+1\over 2}, {b\over 2}, {c\over 2}) \text{SC}({a-1\over 2}, {b\over 2} , {c\over 2})\) for \(a\) odd and \(b,c\) even. The weight is defined as follows. A self-complementary plane partition contains exactly one half of each orbit under the mapping \((i,j,k)\mapsto (a+1-i, b+1-j, c+1-k)\). Define a move from one plane partition into another by removing one half of an orbit and adding the other half. A relative sign is defined, if we fix a plane partition and consider if odd or even number of moves are needed to reach \(P\). \(n(P)\) is 1 or 0 accordingly. The proof translates the problem into counting non-intersecting lattice paths, and then evaluates determinants and Pfaffians.