{"id":"e-2ff0affdc4d5d50d","operation":{"map":"A049020","operation":"basis_map"},"operation_detail":"A049020","conclusion_id":"n-c15181c2ebe13c1d","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":"f8e3649361ce95670af2a508cf6a563f694912f2bde832121c477732c2966d96","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":7541,"certificate_compressed_size":1498,"conclusion_name":"Triangle of numbers a(n,k), 0 <= k <= n: number of set partitions of {1,2,...,n} in which exactly k of the blocks have been distinguished.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}}],"premises":[{"position":0,"node_id":"n-6fea5aff4f44fa54","role":"sequence","name":"a(n) = 1 if n is of the form m(m+3)/2, otherwise 0.","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-c15181c2ebe13c1d","claim":"real_rooted"},{"node_id":"n-c15181c2ebe13c1d","claim":"one_signed_coefficients"},{"node_id":"n-c15181c2ebe13c1d","claim":"hadamard_kernel"}]}