Ehrenborg, Richard The Hankel determinant of exponential polynomials. (English) Zbl 0985.15006 Am. Math. Mon. 107, No. 6, 557-560 (2000). The formulae \(e_n(x)=\sum_{\pi} x^{|\pi|},\) where \(\pi\) ranges over all partitions of \(\{1,\ldots, n\}\) and \(|\pi|\) is the number of blocks of \(\pi\), define the so called exponential polynomials. A proof of C. Radoux’s result [Bull. Soc. Math. Belg. Sér. B 31, 49-55 (1979; Zbl 0439.10006)] that \(\text{det}(e_{i+j}(x))_{0\leq i,j\leq n}= x^{(n+1)n/2} \cdot \prod_{i=0}^n i! \) is given. Unlike Radoux, who uses a functional identity, the present proof is combinatorial in nature. The key idea is that of constructing a certain involution. As a bonus the author obtains the precisely same formula also for the determinants formed from \(e_n^{[\leq 2]}, e_n^{[\geq 2]},e_n^{[=2]},\) where the super ‘indices’ indicate restriction to partitions with block sizes\(\leq 2\), \(=\)2, \(\geq 2\) respectively. As an exercise in using his methods, the author leaves the proof of another result by C. Radoux [Eur. J. Comb. 12, No. 4, 327-329 (1991; Zbl 0805.05003)] for the reader. Reviewer: Alexander Kovačec (Coimbra) Cited in 3 ReviewsCited in 49 Documents MSC: 15A15 Determinants, permanents, traces, other special matrix functions 15B57 Hermitian, skew-Hermitian, and related matrices 05A19 Combinatorial identities, bijective combinatorics Keywords:exponential polynomials; Hankel determinant; involution; bijective proof Citations:Zbl 0439.10006; Zbl 0805.05003 PDFBibTeX XMLCite \textit{R. Ehrenborg}, Am. Math. Mon. 107, No. 6, 557--560 (2000; Zbl 0985.15006) Full Text: DOI Online Encyclopedia of Integer Sequences: Superfactorials: product of first n factorials. a(n) = Product_{i=0..n} i!^2.