{"id":"e-33c9a461744a57dc","operation":{"operation":"obreschkoff_combination"},"operation_detail":null,"conclusion_id":"n-ffeb6f77baa8bcee","premise_count":2,"theorem":"Hermite--Kakeya--Obreschkoff: real-rooted f, g with interlacing zeros give a zero or real-rooted a f + b g for all real a, b (Obreschkoff 1963; Fisk, Polynomials, roots, and interlacing, Thm 1.43)","lean_route":null,"lean_declarations":[],"certificate_hash":"13112f92adb1b0f2fed3b6d13a1adb1d6a4299f3f17c7bbe115ce9504f6f1530","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"obreschkoff_combination","certificate_raw_size":19708,"certificate_compressed_size":3887,"conclusion_name":"obreschkoff_combination of Triangle read by rows: T(n,k) is the number of permutations of 1..n arranged in a circle with exactly k descents.","claims":[{"claim":"real_rooted","from_index":2,"region":{}}],"premises":[{"position":0,"node_id":"n-3ffff24fcb4f40a0","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of permutations of 1..n arranged in a circle with exactly k descents.","uses":{"claim":"real_rooted","from_index":2,"region":{}},"uses_text":"Every row P_n, n >= 2, is zero or real-rooted with all roots in (-inf, +inf)"},{"position":1,"node_id":"n-3ffff24fcb4f40a0","role":"sequence","name":"Triangle read by rows: T(n,k) is the number of permutations of 1..n arranged in a circle with exactly k descents.","uses":{"claim":"interlacing","from_index":3,"kind":"weak","orientation":"unordered"},"uses_text":"Rows P_(m-1), P_m interlace (weak, unordered) for every m >= 3"}],"cited_by":[{"node_id":"n-ffeb6f77baa8bcee","claim":"real_rooted"}]}