{"id":"e-2e5c10abc86c264d","operation":{"operation":"parent_provers","route":"root_domain"},"operation_detail":"root_domain","conclusion_id":"n-efe23a76953cf69d","premise_count":0,"theorem":"polytool parent provers, route root_domain: every parent row P_n, n >= 0, is zero or real-rooted with coefficients of one sign","lean_route":null,"lean_declarations":[],"certificate_hash":"b1e0ba573b9011a608fdc9b1c6cea1e5fee2baab00722061972faa0b8b3509aa","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"parent_provers","certificate_raw_size":1158,"certificate_compressed_size":606,"conclusion_name":"Triangle T(n,k), 0 <= k <= n, read by rows, giving coefficients of the polynomial (x+1)(x+2)...(x+n), expanded in increasing powers of x. T(n,k) is also the unsigned Stirling number |s(n+1, k+1)|, denoting the number of permutations on n+1 elements that contain exactly k+1 cycles.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[],"cited_by":[{"node_id":"n-efe23a76953cf69d","claim":"one_signed_coefficients"},{"node_id":"n-efe23a76953cf69d","claim":"hadamard_kernel"},{"node_id":"n-efe23a76953cf69d","claim":"real_rooted"}]}