{"id":"e-ce077fbc42f0c51e","operation":{"map":"A013611","operation":"basis_map"},"operation_detail":"A013611","conclusion_id":"n-f5f6d7397eb76ee7","premise_count":1,"theorem":"geometric family P_k = g(t) (a + d t)^k: T(p) = g(t) p(a + d t), so the roots of T(p) are those of g and the points (r - a)/d for the roots r of p (elementary)","lean_route":null,"lean_declarations":[],"certificate_hash":"414d2c45c8d20c57de3c6c6a95f835214a6f6cc6d4ed48f9ff90e3f276a801a4","verification_polytool":"replayed","verification_lean":"not_attempted","certificate_kind":"basis_map","certificate_raw_size":15119,"certificate_compressed_size":2793,"conclusion_name":"basis_map_A013611 of Number of compositions of n with k parts, T(n, k) = binomial(n-1, k-1) for n, k >= 1 and T(n, 0) = 0^n, triangle read by rows for n >= 0 and 0 <= k <= n.","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},{"claim":"interlacing","from_index":1,"kind":"weak","orientation":"unordered"}],"premises":[{"position":0,"node_id":"n-3783f6df357ea675","role":"sequence","name":"Number of compositions of n with k parts, T(n, k) = binomial(n-1, k-1) for n, k >= 1 and T(n, 0) = 0^n, triangle read by rows for n >= 0 and 0 <= k <= n.","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-f5f6d7397eb76ee7","claim":"one_signed_coefficients"},{"node_id":"n-f5f6d7397eb76ee7","claim":"hadamard_kernel"},{"node_id":"n-f5f6d7397eb76ee7","claim":"interlacing"},{"node_id":"n-f5f6d7397eb76ee7","claim":"real_rooted"}]}