{"id":"e-1b48038f0f313626","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-f3352d3240f9f08b","premise_count":1,"theorem":"ray basis maps: for L f = (a + b t) f + kappa (t - h) f', b > 0, kappa >= 0, and g real-rooted with every root <= h, of multiplicity m at h, T(u^k) = L^k g sends every real polynomial with all roots in S to one with all roots in (-inf, h] if and only if S lies in (-inf, a + b h + kappa m] (RealRooted rayBasisMap_maps_realRootedIn_iff; the step is Ma and Wang, EJC 15 (2008) R17, Theorem 2)","lean_route":null,"lean_declarations":[],"certificate_hash":"2556c1d68bc82730c2a710580b96c124bc2c6cdfd8e40174e1bc2b47ed655564","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":9093,"certificate_compressed_size":1957,"conclusion_name":"Triangle of unsigned 2-Lah numbers.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-1b45c74901cc1e95","role":"sequence","name":"basis_map_A007318 of Unsigned 2-Stirling numbers of the first kind.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]"}],"cited_by":[{"node_id":"n-f3352d3240f9f08b","claim":"real_rooted"},{"node_id":"n-f3352d3240f9f08b","claim":"one_signed_coefficients"},{"node_id":"n-f3352d3240f9f08b","claim":"hadamard_kernel"}]}