{"id":"e-b0244a114d676405","operation":{"operation":"theorem_check","theorem":"iterated_preserver"},"operation_detail":"iterated_preserver","conclusion_id":"n-a21fa1ea54a634a7","premise_count":0,"theorem":"Iterated real-rootedness preservers (induction on n): if P_m is nonzero and real-rooted with every root in a closed region R and, for every n > m, P_n = a_n(x) S_n(P_(n-1)) where S_n maps every polynomial that is zero or real-rooted with every root in R to zero or a real-rooted polynomial with every root in R and a_n is real-rooted with every root in R, then every P_n is real-rooted with every root in R (the roots of a product are the roots of its factors); an exact degree step keeps every row nonzero; here S_n is the identity","lean_route":null,"lean_declarations":[],"certificate_hash":"98509f1607ae745d7c31e5bf7c36c33b174ebb344b3b670b6197193b5f3d4b36","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"theorem_check","certificate_raw_size":4791,"certificate_compressed_size":1476,"conclusion_name":"Triangle read by rows: T(n,k) is the number of stretching pairs in all permutations in S_{n,k} (=set of permutations in S_n with k cycles) (n >= 3; 1 <= k <= n-2).","claims":[{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"previous_interlaces_next"}],"premises":[],"cited_by":[{"node_id":"n-a21fa1ea54a634a7","claim":"real_rooted"},{"node_id":"n-a21fa1ea54a634a7","claim":"one_signed_coefficients"},{"node_id":"n-a21fa1ea54a634a7","claim":"hadamard_kernel"},{"node_id":"n-a21fa1ea54a634a7","claim":"interlacing"}]}