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The Hankel determinant of exponential polynomials. (English) Zbl 0985.15006
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.

MSC:
15A15 Determinants, permanents, traces, other special matrix functions
15B57 Hermitian, skew-Hermitian, and related matrices
05A19 Combinatorial identities, bijective combinatorics
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