On the spiral property of the q-derangement numbers

✍ Scribed by Xiang-De Zhang

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 136 KB
• Volume: 159
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/0012-365x(95)00090-j

✦ Synopsis

We discover the spiral property of the q-derangement numbers arising from q-counting of derangements by the major index. The spiral property implies that the polynomial is unimodal and the maximum coefficient appears exactly in the middle, which confirms a conjecture of Chen and Rota.

Let Dn(q) denote the q-derangement numbers for the major index; Dn(q) has the following expression [1,3,4]:

Chen and Rota [1] have shown the unimodality of Dn(q) and posed the following conjecture: the maximum coefficient of Dn(q) appears in the middle of the polynomial, namely the coefficient of q Fn(n-1)/4], where rx~ is the usual notation for the smallest integer not less than x. The polynomials Dn(q) are listed in [1] for n ~< 8: Dl(q) = 0, Oz(q) = q, D3(q) = q + q2, D4(q) = q + 2q 2 + 2q 3 + 2q 4 + q5 + q6, Ds(q) = q + 3q 2 + 5q 3 + 7q 4 q-8q 5 + 8q 6 -q-6q 7 + 4q 8 + 2q 9, D6(q)=q+4q 2+9q 3+ 16q 4+24q 5+32q 6+37q 7+38q 8+35q 9 + 28qlO + 20qll + 12q12 + 6q13 + 2q14 + q15, 1 Supported by the Nature Science Foundation of China.