{"id":"e-0fa0df960e50bc7a","operation":{"operation":"theorem_check","theorem":"ma_wang_interval"},"operation_detail":"ma_wang_interval","conclusion_id":"n-2cabfd060b04f15d","premise_count":0,"theorem":"S.-M. Ma and Y. Wang, q-Eulerian polynomials and polynomials with only real zeros, Electron. J. Combin. 15 (2008) R17, Theorem 2 and Corollary 3 (arXiv:math/0611826), in the closed two-endpoint form of the RealRooted library (https://github.com/PerAlexandersson/RealRooted, RealRooted.affineEndpointStep_closed), after the change of variable x = U + (U - L) y, with the endpoint multiplicities tracked exactly through the induction","lean_route":null,"lean_declarations":["RealRooted.affineEndpointStep_closed","RealRooted.affineEndpointStep_roots_mem_Icc_iff","RealRooted.affineEndpointStep_rootMultiplicity_zero","RealRooted.affineEndpointStep_rootMultiplicity_neg_one","RealRooted.MaWang.endpointStep","RealRooted.MaWang.affineEndpointSequence","RealRooted.normalizedAffineEndpointSequence_roots_mem_Icc_iff"],"certificate_hash":"6fda967c76d614a03d5be1a9dd819e3ba8e08bb5627d483df8668ed2d2ab053c","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"theorem_check","certificate_raw_size":3963,"certificate_compressed_size":1311,"conclusion_name":"Triangle of numbers T(n,k) = k!*Stirling2(n,k) = A000142(k)*A048993(n,k) read by rows, T(n, k) for 0 <= k <= n.","claims":[{"claim":"real_rooted","from_index":1,"region":{"lower":{"closed":false,"value":"-1"},"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[],"cited_by":[{"node_id":"n-2cabfd060b04f15d","claim":"real_rooted"},{"node_id":"n-2cabfd060b04f15d","claim":"one_signed_coefficients"},{"node_id":"n-2cabfd060b04f15d","claim":"hadamard_kernel"},{"node_id":"n-2cabfd060b04f15d","claim":"interlacing"}]}