{"id":"n-5e3f8ad0b0a35d3e","revision":1,"first_index":0,"fingerprint":"f6-efa00b2a69da6c9b","name":"Triangle read by rows: T(n,k) is the number of permutations of [n] with k increasing runs of length at least 2.","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"half_affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":false,"rows":[["1"],["1"],["1","1"],["1","5"],["1","18","5"],["1","58","61"],["1","179","479","61"],["1","543","3111","1385"],["1","1636","18270","19028","1385"],["1","4916","101166","206276","50521"],["1","14757","540242","1949762","1073517","50521"],["1","44281","2819266","16889786","17460701","2702765"],["1","132854","14494859","137963364","241595239","82112518","2702765"],["1","398574","73802835","1081702420","3002137335","1869618654","199360981"],["1","1195735","373398489","8236142455","34591152955","35576491869","8200548715","199360981"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","1","2","6","24","120","720","5040","40320","362880","3628800","39916800","479001600","6227020800","87178291200"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A008971","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["1","-1"],["0","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["0","2","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":159972,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-4c46966ead0b055f"]},{"id":159973,"position":1,"claim":"one_signed_coefficients","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":0},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 0: the coefficients of P_n have one sign","region_exact":null,"edges":["e-4c46966ead0b055f"]},{"id":159974,"position":2,"claim":"hadamard_kernel","from_index":0,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":0},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 0, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-4c46966ead0b055f"]}],"region_exact":{},"incoming_edges":[{"id":"e-4c46966ead0b055f","operation":{"index_shift":0,"operation":"gamma_polynomial","scale":"1/4"},"operation_detail":"scale 1/4, shift 0","conclusion_id":"n-5e3f8ad0b0a35d3e","premise_count":1,"theorem":"gamma criterion (RealRooted IsGammaExpansion.isRealRooted_and_hasRootsNonpos_iff): if P = sum_j gamma_j x^j (1 + x)^(d - 2j) with deg Gamma <= d/2, then P is nonzero and real-rooted with roots in (-inf, 0] if and only if Gamma(z) = sum_j gamma_j z^j is; Gamma(c z) with c != 0 has the roots of Gamma divided by c","lean_route":null,"lean_declarations":[],"certificate_hash":"5b3e4033348c4d712d4422010d63f6ad0e8ad6805080080beb4af51b62151a0e","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"0"}}}],"premises":[{"position":0,"node_id":"n-d8312bb744732845","role":"sequence","name":"type_b_eulerian","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]"}]}],"used_by":[{"id":"e-0b45882766291d5a","operation":"preserver_image","operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-0922e9490abc04e5","conclusion_name":"one_plus_derivative of Triangle read by rows: T(n,k) is the number of permutations of [n] with k increasing runs of length at least 2."},{"id":"e-1aee28e48ce84e67","operation":"preserver_image","operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-07eb37e9371d0ecc","conclusion_name":"theta_plus_one_squared of Triangle read by rows: T(n,k) is the number of permutations of [n] with k increasing runs of length at least 2."},{"id":"e-42ac1b21fb8d0456","operation":"preserver_image","operation_detail":"affine, shift 0","conclusion_id":"n-5d7c3f4772a9d8bb","conclusion_name":"shift_x_plus_one of Triangle read by rows: T(n,k) is the number of permutations of [n] with k increasing runs of length at least 2."},{"id":"e-6301aab0e71cd09e","operation":"preserver_image","operation_detail":"derivative_polynomial, shift 0","conclusion_id":"n-add6df1ce75e6a78","conclusion_name":"derivative of Triangle read by rows: T(n,k) is the number of permutations of [n] with k increasing runs of length at least 2."},{"id":"e-b88134e74c47445b","operation":"preserver_image","operation_detail":"euler_polynomial, shift 0","conclusion_id":"n-0d7d4f614fd8e324","conclusion_name":"theta_plus_one of Triangle read by rows: T(n,k) is the number of permutations of [n] with k increasing runs of length at least 2."}],"used_by_total":5,"links":[{"id":"l-1af0a676b52d30ee","kind":"oeis_match","node_a_id":"n-5e3f8ad0b0a35d3e","node_b_id":null,"fingerprint":null,"oeis_id":"A159333","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":0,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":-2,"oeis_term_position":2,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A159333","oeis_nonn":false},"other_id":null,"other":null},{"id":"l-305cbd8350db2550","kind":"oeis_match","node_a_id":"n-5e3f8ad0b0a35d3e","node_b_id":null,"fingerprint":null,"oeis_id":"A370360","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":1,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":0,"target":"row_sums","terms_compared":14},"in_catalog":false,"oeis_id":"A370360","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-4914ee4f6d5f49c9","kind":"oeis_match","node_a_id":"n-5e3f8ad0b0a35d3e","node_b_id":null,"fingerprint":null,"oeis_id":"A354074","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":3,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":1,"target":"row_sums","terms_compared":12},"in_catalog":false,"oeis_id":"A354074","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-5e130251933098bd","kind":"oeis_match","node_a_id":"n-5e3f8ad0b0a35d3e","node_b_id":null,"fingerprint":null,"oeis_id":"A000142","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A000142","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-8053612d7fa13774","kind":"oeis_match","node_a_id":"n-5e3f8ad0b0a35d3e","node_b_id":null,"fingerprint":null,"oeis_id":"A104150","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":1,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A104150","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-9e4cf74770e8a9cf","kind":"oeis_match","node_a_id":"n-5e3f8ad0b0a35d3e","node_b_id":null,"fingerprint":null,"oeis_id":"A124355","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":2,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":2,"target":"row_sums","terms_compared":13},"in_catalog":false,"oeis_id":"A124355","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-9fe0f6d1ee42aa5d","kind":"oeis_match","node_a_id":"n-5e3f8ad0b0a35d3e","node_b_id":null,"fingerprint":null,"oeis_id":"A358185","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":0,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":2,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A358185","oeis_nonn":false},"other_id":null,"other":null},{"id":"l-b1faf2607e6768ef","kind":"oeis_match","node_a_id":"n-5e3f8ad0b0a35d3e","node_b_id":null,"fingerprint":null,"oeis_id":"A338112","in_catalog":0,"target":"row_sums","row_alignment":null,"rows_matched":null,"oeis_nonn":1,"details":{"alignment":{"first_row":3,"low_zeros_dropped":false,"oeis_offset":1,"oeis_term_position":2,"target":"row_sums","terms_compared":12},"in_catalog":false,"oeis_id":"A338112","oeis_nonn":true},"other_id":null,"other":null},{"id":"l-e749be7a7b9124a1","kind":"oeis_match","node_a_id":"n-5e3f8ad0b0a35d3e","node_b_id":null,"fingerprint":null,"oeis_id":"A008971","in_catalog":1,"target":"triangle","row_alignment":"node_lengths","rows_matched":12,"oeis_nonn":1,"details":{"alignment":{"first_row":0,"low_zeros_dropped":false,"oeis_offset":0,"oeis_term_position":0,"row_alignment":"node_lengths","rows_matched":12,"target":"triangle","terms_compared":45},"in_catalog":true,"oeis_id":"A008971","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A008971","relation":"identification"},{"oeis_id":"A000142","relation":"match_row_sums"},{"oeis_id":"A104150","relation":"match_row_sums"},{"oeis_id":"A124355","relation":"match_row_sums"},{"oeis_id":"A159333","relation":"match_row_sums"},{"oeis_id":"A338112","relation":"match_row_sums"},{"oeis_id":"A354074","relation":"match_row_sums"},{"oeis_id":"A358185","relation":"match_row_sums"},{"oeis_id":"A370360","relation":"match_row_sums"},{"oeis_id":"A008971","relation":"match_triangle"}],"bfiles":[{"node_id":"n-5e3f8ad0b0a35d3e","oeis_id":"A000142","target":"row_sums","oeis_offset":0,"first_row":0,"rows_used":200,"terms_matched":15,"oeis_terms_agreeing":23,"oeis_terms_available":23,"term_count":200,"raw_size":34064},{"node_id":"n-5e3f8ad0b0a35d3e","oeis_id":"A008971","target":"triangle","oeis_offset":0,"first_row":0,"rows_used":62,"terms_matched":45,"oeis_terms_agreeing":45,"oeis_terms_available":45,"term_count":992,"raw_size":46446}],"oeis_texts":[{"oeis_id":"","target":"node","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0: 1;\n  1: 1;\n  2: 1,    1;\n  3: 1,    5;\n  4: 1,   18,      5;\n  5: 1,   58,     61;\n  6: 1,  179,    479,     61;\n  7: 1,  543,   3111,   1385;\n  8: 1, 1636,  18270,  19028,  1385;\n  9: 1, 4916, 101166, 206276, 50521;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+(n-1)*x)*P(n-1,x) + (2*x-2*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(1 + (n - 1)*x)*P[n - 1, x] + (2*x - 2*x^2)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (1 + (n - 1)*x)*v[i - 1] + (2*x - 2*x^2)*deriv(v[i - 1], x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}},{"oeis_id":"A008971","target":"triangle","first_row":0,"oeis_offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false,"rows_shown":10,"polytool_commit":"c48629374597092cc41ef4d6273f37991516f984","triangle_example":"Triangle begins:\n  0: 1;\n  1: 1;\n  2: 1,    1;\n  3: 1,    5;\n  4: 1,   18,      5;\n  5: 1,   58,     61;\n  6: 1,  179,    479,     61;\n  7: 1,  543,   3111,   1385;\n  8: 1, 1636,  18270,  19028,  1385;\n  9: 1, 4916, 101166, 206276, 50521;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (1+(n-1)*x)*P(n-1,x) + (2*x-2*x^2)*d/dx P(n-1,x) for n >= 1, with P(0,x) = 1.","mathematica":"P[0, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[(1 + (n - 1)*x)*P[n - 1, x] + (2*x - 2*x^2)*D[P[n - 1, x], x]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; for(n = 1, N, my(i = n + 1); v[i] = (1 + (n - 1)*x)*v[i - 1] + (2*x - 2*x^2)*deriv(v[i - 1], x)); v};\nrow(n, p) = {my(c = Vecrev(p)); c};\nT(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};\nT(9)","reading":{"first_row":0,"offset":0,"reversed":false,"trim_zeros":"none","regular_triangle":false}}],"derivation":{"root":"n-d8312bb744732845","steps":["\u03b3 (scale 1/4)"],"edges":["e-4c46966ead0b055f"],"text":"type_b_eulerian \u2192 \u03b3 (scale 1/4)"},"headline":"A008971: Triangle read by rows: T(n,k) is the number of permutations of [n] with k increasing runs of length at least 2.","chain":"type_b_eulerian \u2192 \u03b3 (scale 1/4)"}
