Logarithmically concave sequence

The rows of Pascal's triangle are examples for logarithmically concave sequences.

In mathematics, a sequence a = (a₀, a₁, ..., a_n) of nonnegative real numbers is called a logarithmically concave sequence, or a log-concave sequence for short, if a_i² ≥ a_i−1a_i+1 holds for 0 < i < n .

Remark: some authors (explicitly or not) add two further conditions in the definition of log-concave sequences:

• a is non-negative
• a has no internal zeros; in other words, the support of a is an interval of Z.

These conditions mirror the ones required for log-concave functions.

Sequences that fulfill the three conditions are also called Pólya Frequency sequences of order 2 (PF₂ sequences). Refer to chapter 2 of for a discussion on the two notions. For instance, the sequence (1,1,0,0,1) satisfies the concavity inequalities but not the internal zeros condition.

Examples of log-concave sequences are given by the binomial coefficients along any row of Pascal's triangle and the elementary symmetric means of a finite sequence of real numbers.

References

• Stanley, R. P. (December 1989). "Log-Concave and Unimodal Sequences in Algebra, Combinatorics, and Geometry". Annals of the New York Academy of Sciences. 576 (1): 500–535. Bibcode:1989NYASA.576..500S. doi:10.1111/j.1749-6632.1989.tb16434.x.

See also

• Unimodality
• Logarithmically concave function
• Logarithmically concave measure