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Let $n \times n$ matrix $M(n):=(m_{ij})$ be defined by $$m_{ij} = \min(i,j)$$

This matrix enumerates certain combinatorial objects. Do you know any interesting properties of matrix $M$?

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    $\begingroup$ Could you make precise the motivation ? which sort of combinatorial objects are enumerated ? $\endgroup$ Aug 3, 2015 at 10:24
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    $\begingroup$ This is a very interesting matrix (Brownian bridge kernel); for instance, see section 3 of my notes: arxiv.org/pdf/1411.4107v2.pdf $\endgroup$
    – Suvrit
    Aug 3, 2015 at 14:06
  • $\begingroup$ Thanks for the information. The enumeration is related with the counting of certain vertex-weighted graph. $\endgroup$
    – hkju
    Aug 3, 2015 at 19:43
  • $\begingroup$ See arxiv:math/0606163 by Bona, Ju and Yoshida. $\endgroup$
    – hkju
    Aug 3, 2015 at 19:52

1 Answer 1

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Let $A=(a_{ij})$ be the upper triangular matrix such that $a_{ij}= \begin{cases} 1, \textrm{if $i \leq j$}\\ 0, \textrm{else} \end{cases}$, then the matrix $M$ is equal to the product $A^{T}A$. Many interesting properties of $M$ follow from this.

For example, one concludes that $M$ has determinant $1$, and is positive definite. What else do you want to know?

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