From then on, Nicomachus’s table of figurate numbers, expanded with additional rows and columns, appeared in numerous arithmetic treatises, including those of al-Baghdadi (c. 980–1037), Avicenna and Alhazen. Although a few new results were proved, it was in the 13th century that the understanding of figurate numbers changed radically.
Toward a combinatorial interpretation
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At the same time, two mathematicians working far apart—al-Farisi in Persia and al-Banna in Morocco, more precisely in Marrakesh—challenged the prevailing understanding of figurate numbers. Both linked combinatorial questions to figurate numbers.
Al-Banna was particularly interested in a lexicographical question:
"The sum of squares is useful for forming triliteral words, counting the vocabulary of languages, and the like. For example: How many triliteral words can be formed from the letters of the alphabet without counting permutations separately?
