.. _sphx_glr_api_gallery_combinatorics_special_numbers:

Special numbers
-----------------

Stirling numbers, Bell numbers, and Catalan numbers.


.. raw:: html

  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


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    <div class="sphx-glr-thumbnails">

.. thumbnail-parent-div-open

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    <div class="sphx-glr-thumbcontainer" tooltip="Checks by brute force that Stirling numbers of the first kind count permutations by their number of cycles and that Stirling numbers of the second kind count partitions of a labeled set into k blocks. It then recovers Stirling&#x27;s own use of them, converting ordinary powers into falling factorials, and sums the second kind to get the Bell numbers.">

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  .. image:: /api/gallery/combinatorics/special_numbers/images/thumb/sphx_glr_plot_01_stirling_numbers_thumb.png
    :alt:

  :doc:`/api/gallery/combinatorics/special_numbers/plot_01_stirling_numbers`

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      <div class="sphx-glr-thumbnail-title">Stirling's numbers: cycles, set partitions, and Bell numbers</div>
    </div>


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    <div class="sphx-glr-thumbcontainer" tooltip="Draws all 14 triangulations of a convex hexagon, the problem Euler posed to Goldbach in 1751, and checks that the same Catalan numbers count balanced parenthesizations, Catalan&#x27;s bracketing problem of 1838.">

.. only:: html

  .. image:: /api/gallery/combinatorics/special_numbers/images/thumb/sphx_glr_plot_02_catalan_numbers_thumb.png
    :alt:

  :doc:`/api/gallery/combinatorics/special_numbers/plot_02_catalan_numbers`

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      <div class="sphx-glr-thumbnail-title">Catalan numbers: Euler's polygon triangulations and Catalan's brackets</div>
    </div>


.. thumbnail-parent-div-close

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    </div>


.. toctree::
   :hidden:

   /api/gallery/combinatorics/special_numbers/plot_01_stirling_numbers
   /api/gallery/combinatorics/special_numbers/plot_02_catalan_numbers