### Description

Going to Airflow web UI, Browse>DAG Run displays the list of runs, but there is no way to search or filter based on the text in the "Note" column.

### Use case/motivation

It is possible to do a free text search for the "Run Id" field. The Note field may contain pieces of information that may be relevant to find, or to filter on the basis of these notes.

### Related issues

Sorting by Note in List Dag Run fails: 
https://github.com/apache/airflow/issues/30041

### Are you willing to submit a PR?

- [ ] Yes I am willing to submit a PR!

### Code of Conduct

- [X] I agree to follow this project's [Code of Conduct](https://github.com/apache/airflow/blob/main/CODE_OF_CONDUCT.md)

current_page,
    num_of_pages,
    search=None,
    status=None,
    tags=None,
    window=7,
    sorting_key=None,
    sorting_direction=None,
):
    """
    Generates the HTML for a paging component using a similar logic to the paging
    auto-generated by Flask managed views. The paging component defines a number of
    pages visible in the pager (window) and once the user goes to a page beyond the
    largest visible, it would scroll to the right the page numbers and keeps the
    current one in the middle of the pager component. When in the last pages,
    the pages won't scroll and just keep moving until the last page. Pager also contains
    <first, previous, ..., next, last> pages.
    This component takes into account custom parameters such as search, status, and tags
    which could be added to the pages link in order to maintain the state between
    client and server. It also allows to make a bookmark on a specific paging state.
    :param current_page: the current page number, 0-indexed
    :param num_of_pages: the total number of pages
    :param search: the search query string, if any
    :param status: 'all', 'active', or 'paused'
    :param tags: array of strings of the current filtered tags
    :param window: the number of pages to be shown in the paging component (7 default)
    :param sorting_key: the sorting key selected for dags, None indicates that sorting is not needed/provided
    :param sorting_direction: direction of sorting, 'asc' or 'desc',