{"id":1196,"date":"2019-02-06T10:06:17","date_gmt":"2019-02-06T18:06:17","guid":{"rendered":"https:\/\/colleges.claremont.edu\/ccms\/?post_type=tribe_events&#038;p=1196"},"modified":"2019-03-25T21:25:03","modified_gmt":"2019-03-26T04:25:03","slug":"antc-talk-curtis-bennett-csulb","status":"publish","type":"tribe_events","link":"https:\/\/colleges.claremont.edu\/ccms\/event\/antc-talk-curtis-bennett-csulb\/","title":{"rendered":"Fibonacci and Lucas analogues of binomial coefficients and what they count (Curtis Bennett, CSULB)"},"content":{"rendered":"<p>A Fibonomial is what is obtained when you replace each term of the binomial coefficients $ {n \\choose k}$ by the corresponding Fibonacci number.<span class=\"Apple-converted-space\">\u00a0 <\/span>For example, the Fibonomial<\/p>\n<p>$${ 6\\brace 3 } = \\frac{F_6 \\cdot F_5 \\cdot \\dots \\cdot F_1}{(F_3\\cdot F_2 \\cdot F_1)(F_3\\cdot F_2 \\cdot F_1)} = \\frac{8\\cdot5\\cdot3\\cdot2\\cdot1\\cdot1}{(2\\cdot1\\cdot1)(2\\cdot1\\cdot1)} = 60$$<\/p>\n<p>since the first six Fibonacci numbers are 1, 1, 2, 2, 5, and 8.<span class=\"Apple-converted-space\">\u00a0 <\/span>Curiously the Fibonomials are always integers, raising the combinatorial question:<span class=\"Apple-converted-space\">\u00a0 w<\/span>hat do they count?<span class=\"Apple-converted-space\">\u00a0 <\/span>In this talk we introduce and provide a little history of the Fibonomials.<span class=\"Apple-converted-space\">\u00a0 <\/span>We then provide a simple object the Fibonomials enumerate.<span class=\"Apple-converted-space\">\u00a0 <\/span>We will use this new object to prove various Fibonomial analogues of standard identities on binomial coefficients and discuss further generalizations including the Lucanomials.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>A Fibonomial is what is obtained when you replace each term of the binomial coefficients $ {n \\choose k}$ by the corresponding Fibonacci number.\u00a0 For example, the Fibonomial $${ 6\\brace [&hellip;]<\/p>\n","protected":false},"author":73,"featured_media":0,"template":"","meta":{"_acf_changed":false,"_price":"","_stock":"","_tribe_ticket_header":"","_tribe_default_ticket_provider":"","_tribe_ticket_capacity":"0","_ticket_start_date":"","_ticket_end_date":"","_tribe_ticket_show_description":"","_tribe_ticket_show_not_going":false,"_tribe_ticket_use_global_stock":"","_tribe_ticket_global_stock_level":"","_global_stock_mode":"","_global_stock_cap":"","_tribe_rsvp_for_event":"","_tribe_ticket_going_count":"","_tribe_ticket_not_going_count":"","_tribe_tickets_list":"[]","_tribe_ticket_has_attendee_info_fields":false,"_tribe_events_status":"","_tribe_events_status_reason":"","_tribe_events_is_hybrid":"","_tribe_events_is_virtual":"","_tribe_events_virtual_video_source":"","_tribe_events_virtual_embed_video":"","_tribe_events_virtual_linked_button_text":"","_tribe_events_virtual_linked_button":"","_tribe_events_virtual_show_embed_at":"","_tribe_events_virtual_show_embed_to":[],"_tribe_events_virtual_show_on_event":"","_tribe_events_virtual_show_on_views":"","_tribe_events_virtual_url":"","footnotes":"","_tec_slr_enabled":"","_tec_slr_layout":""},"tags":[],"tribe_events_cat":[13],"class_list":["post-1196","tribe_events","type-tribe_events","status-publish","hentry","tribe_events_cat-antc-seminar","cat_antc-seminar"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.2 - 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