Editing Hermite polynomials (section)

===Combinatorial interpretation of coefficients===
In the Hermite polynomial {{math|''He''<sub>''n''</sub>(''x'')}} of variance 1, the absolute value of the coefficient of {{math|''x''<sup>''k''</sup>}} is the number of (unordered) partitions of an {{mvar|n}}-element set into {{mvar|k}} singletons and {{math|{{sfrac|''n'' − ''k''|2}}}} (unordered) pairs. Equivalently, it is the number of involutions of an {{mvar|n}}-element set with precisely {{mvar|k}} fixed points, or in other words, the number of matchings in the [[complete graph]] on {{mvar|n}} vertices that leave {{mvar|k}} vertices uncovered (indeed, the Hermite polynomials are the [[matching polynomial]]s of these graphs). The sum of the absolute values of the coefficients gives the total number of partitions into singletons and pairs, the so-called [[Telephone number (mathematics)|telephone numbers]]
: 1, 1, 2, 4, 10, 26, 76, 232, 764, 2620, 9496,... {{OEIS|A000085}}.

This combinatorial interpretation can be related to complete exponential [[Bell polynomials]] as
<math display="block">\operatorname{He}_n(x) = B_n(x, -1, 0, \ldots, 0),</math>
where {{math|1=''x''<sub>''i''</sub> = 0}} for all {{math|''i'' > 2}}.

These numbers may also be expressed as a special value of the Hermite polynomials:<ref name="gfgt">{{citation
 | last1 = Banderier | first1 = Cyril
 | last2 = Bousquet-Mélou | first2 = Mireille | author2-link = Mireille Bousquet-Mélou
 | last3 = Denise | first3 = Alain
 | last4 = Flajolet | first4 = Philippe | author4-link = Philippe Flajolet
 | last5 = Gardy | first5 = Danièle
 | last6 = Gouyou-Beauchamps | first6 = Dominique
 | arxiv = math/0411250 | doi = 10.1016/S0012-365X(01)00250-3 | issue = 1–3 | journal = [[Discrete Mathematics (journal)|Discrete Mathematics]] | mr = 1884885 | pages = 29–55 | title = Generating functions for generating trees | volume = 246 | year = 2002| s2cid = 14804110
 }}</ref>
<math display="block">T(n) = \frac{\operatorname{He}_n(i)}{i^n}.</math>