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=Dyck path=
A [[lattice path]] on the square lattice from the origin $(0,0)$ to some point $(n,n)$ consisting of $2n$ steps of the form $N : (x,y) \rightarrow (x,y+1)$ and $E : (x,y) \rightarrow (x+1,y)$ with the property that the path
never passes below the line $y=x$.

The number of Dyck paths of length $2n$ is given by the $n$-th [[Catalan number]]
$$
C_n = \frac{1}{n+1}\binom{2n}{n} \ .
$$

====References====

=Catalan number=
The $n$-th Catalan number
$$
C_n = \frac{1}{n+1}\binom{2n}{n} \ .
$$
The [[generating function]] is given by
$$
\sum_{n=1}^\infty C_n z^n = \frac{1-\sqrt{1-4z}}{2z} \ .
$$
The Catalan numbers appear in the enumeration of a number of combinatorially defined object:
* [[Bernoulli excursion]]
* [[Dyck path]]s
* Parenthesised sequences; words of the [[Dyck language]]
* Complete [[Binary tree|binary]] [[rooted plane tree]]s

====References====

=Poisson ratio=
The ratio of longitudinal extension to lateral compression when an elastic substance is put under tension.

See: [[Elasticity, mathematical theory of]]; [[Lamé constants]].

====References====
* Horace Lamb, "Statics", Cambridge University Press (1960)

=Elastic modulus=
''Young's modulus''

The ratio of longitudinal extension to force applied per unit area when an elastic substance is put under tension.

See: [[Elasticity, mathematical theory of]]; [[Lamé constants]].

====References====
* Horace Lamb, "Statics", Cambridge University Press (1960)

=Partition symbol=
A notation used to compactly express propositions of [[partition calculus]].  The symbol
$$
\alpha \rightarrow (\beta)_\gamma^r
$$
for cardinals $\alpha,\beta,\gamma$ and natural number $r$, denotes the following proposition.

Given a set $S$ and a colouring of $S^r$ into a set of $\gamma$ colours, there exists a subset $T$ of $S$ of cardinality $|T|=\beta$ such that the colouring restricted to $T^r$ is monochrome.

Here a colouring of a set $X$ by a set of colours $C$ is simply a partition of $X$ into parts indexed by the set $C$.

The symbol
$$
\alpha \rightarrow (\beta_1,\ldots,\beta_j)^r
$$
denotes the following proposition:

Given a set $S$ of cardinality $\alpha$ and a colouring of $S^r$ by $j$ colours, there exists an index $i$ subset $T$ of $S$ of cardinality $|T|=\beta_i$ such that the colouring restricted to $T^r$ is monochrome.

Examples.

* [[Ramsey theorem|Ramsey's theorem]]: $\omega \rightarrow (\omega)_n^r$.
* Sierpinski's theorem: $c \not\rightarrow (\omega_1,\omega_2)^2$.


====References====
* M.E. Rudin, "Lectures on set theoretic topology", Amer. Math. Soc. (1975) ISBN 0-8218-1673-X  {{ZBL|0318.54001}}

=Isthmus=
''bridge, co-loop''

An isthmus of a [[graph]] is an edge for which deletion increases the number of connected components of the graph.  

An element of a [[matroid]] $M$ on a set $E$ is an element of $E$ which is in every basis for $M$.  An element of $E$ is a co-loop of $M$ if and only if it is a ''loop'' of the [[dual matroid]] $M^*$, that is, does not belong to any base of $M^*$.  If $M$ is a [[graphic matroid]], then the definitions coincide.

====References====
* J. G. Oxley, "Matroid Theory" (2 ed) Oxford University Press (2011) ISBN 978-0-19-856694-6  {{ZBL|1254.05002}}
* D. J. A. Welsh, "Matroid Theory", Dover (2010) [1976] ISBN 0486474399 {{ZBL|}}0343.05002