OFFSET
1,3
COMMENTS
A symbolic sequence on n symbols 0, 1, ..., n - 1 is a sequence of length n*(n+1)/2 consisting of n occurrences of 0, (n - 1) occurrences of 1, ..., one occurrence of n-1 and satisfying the condition that between any two consecutive occurrences of the symbol i it has exactly one occurrence of the symbol i+1. For example, the two symbolic sequences on 3 symbols are 012010 and 010210.
The number of symbolic sequences on n symbols is listed in A003121. Anderson shows that out of the possible 12 symbolic sequences on 4 symbols only 10 can be realized as the arrangement of the 10 roots of some real polynomial of degree 4 and its derivatives.
Kostov shows that out of the possible 286 symbolic sequences on 5 symbols only 116 can be realized as the arrangement of the 15 roots of some real polynomial of degree 5 and its derivatives. V. I. Arnold has asked if the limit as n -> infinity of a(n)/A003121(n) is zero.
LINKS
B. Anderson, Polynomial root dragging, Amer. Math. Monthly 100 (1993), 864-866.
V. Kostov, Discriminant sets of families of hyperbolic polynomials of degree 4 and 5, Serdica Math. J. 28 (2002), no.2, 117-152.
B. Shapiro and M. Shapiro, A few riddles behind Rolle's theorem, arXiv:math/0302215 [math.CA], 2003-2005.
EXAMPLE
a(3)=2. Let p be a real polynomial of degree 3 such that the 3 zeros x_1 < x_2 < x_3 of p are real and distinct. Let y_1 and y_2 denote the zeros of p' and let z_1 denote the zero of p''. Then by Rolle's theorem, x_1 < y_1 < x_2 < y_2 < x_3 and y_1 < z_1 < y_2. If z_1 does not coincide with x_2 (the generic case) then there are two possible arrangements for the 6 roots of p and its derivatives: either ( x_1 < y_1 < z_1 < x_2 < y_2 < x_3 ) or ( x_1 < y_1 < x_2 < z_1 < y_2 < x_3 ). These arrangements may be encoded in a symbolic sequence on 3 symbols namely either 012010 or 010210 (replace the x's by 0's, the y's by 1's and the z's by 2's). Both these arrangements are actually realized for some degree 3 polynomial. For example, the six roots of p = x*(x-1)*(x-3/2) and its derivatives are arranged symbolically as 012010 and the six roots of p = x*(x-1)*(x-3) and its derivatives are arranged symbolically as 010210.
CROSSREFS
KEYWORD
nonn,hard,more
AUTHOR
Peter Bala, Jul 18 2007
STATUS
approved