{"id":"combinatorics","domain":"discrete","type":"Concept","title":"Combinatorics","slug":"combinatorics","url":"/mathematics/discrete/combinatorics/","summary":"The mathematics of counting: arrangements, selections, combinations. Essential to probability, cryptography, algorithm analysis.","key_formulas":["P(n,k)=n!/(n-k)! — permutations","C(n,k)=n!/[k!(n-k)!] — combinations"],"added_to_codex":"2026-05-27","last_verified":"2026-05-27","sources":[{"tier":1,"citation":"Stanley, R.P. (2011). Enumerative Combinatorics. Vol. 1, 2nd ed. Cambridge University Press."},{"tier":1,"citation":"Graham, R.L., Knuth, D.E. and Patashnik, O. (1994). Concrete Mathematics. 2nd ed."},{"tier":2,"citation":"Graham, R.L., Rothschild, B.L. and Spencer, J.H. (1990). Ramsey Theory. 2nd ed. Wiley."},{"tier":3,"citation":"Alon, N. and Spencer, J.H. (2016). The Probabilistic Method. 4th ed. Wiley."}],"relationships":[{"type":"used_by","target":"probability","label":"Probability"},{"type":"related_to","target":"graph-theory","label":"Graph Theory"},{"type":"related_to","target":"aryabhatiya","label":"Āryabhaṭīya — combinatorial roots"}],"entry_status":"live","reading_levels_available":["curious","exploring","deep_dive"],"completeness":1.0,"codex_version":"1.8"}