{"id":"e-857a86cbb6c7881e","operation":{"map":"A008277","operation":"basis_map"},"operation_detail":"A008277","conclusion_id":"n-e65f982fa1b436c5","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":"5dbea33f5f1b028325c49d7d9ae2cd84bbfd7f9c5bc20d4f0e3680abe026117a","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":7528,"certificate_compressed_size":1500,"conclusion_name":"Triangle of Stirling numbers of 2nd kind, S(n,k), n >= 0, 0 <= k <= n.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-50902364601606aa","role":"sequence","name":"basis-map coefficients of Triangle of Stirling numbers of 2nd kind, S(n,k), n >= 0, 0 <= k <= n.","uses":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"1"}}},"uses_text":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 1]"}],"cited_by":[{"node_id":"n-e65f982fa1b436c5","claim":"real_rooted"},{"node_id":"n-e65f982fa1b436c5","claim":"one_signed_coefficients"},{"node_id":"n-e65f982fa1b436c5","claim":"hadamard_kernel"}]}