Unusual power products and the ideal [𝑦²]
Kathleen B. O’Keefe · Proceedings of the American Mathematical Society · 1966
A power product P=yi(lyi(2) * yi,n), where Yi(j) is the i(j)th derivative of y, has weight w = =1 i(j) and degree d = n. The sequence of integers (e1, e2, , en), where ek= = i(k) -k(k-1), is called the weight sequence of P. According to a result of H. Levi ([1], Theorem 1.2, p. 545), if some ek 2; however, this note is concerned only with [y2]. All unusual power products with weight sequences consisting of 0's, l's, and 2's are described by Theorem IV, [2]: A power product P with weight sequence (el, * * *, en), 0! e, < 2, i= 1, ***, n, is unusual if and only if somewhere in the sequence at least one of the following patterns appears: 1, 2, 2, 1; 1, 2, 2, 2, 2, 0; 0, 2, 2, 2, 2, 1; 0, 2, 2, 2, 2, 2, 2, 0. For particular arrangements of 0's, l's, 2's, and 3's, similar results may be obtained; for example, using the notation of [2],