{"id":"n-a21fa1ea54a634a7","revision":0,"first_index":0,"fingerprint":"f6-34eaf45657600f7a","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).","polytool_family":null,"interpretation":null,"rows_observed":15,"degree_pattern":"affine","degree_slope":1,"degree_intercept":0,"palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":true,"rows":[[],["1","1"],["10","15","5"],["90","165","90","15"],["840","1750","1225","350","35"],["8400","19180","15750","5950","1050","70"],["90720","222264","204624","92610","22050","2646","126"],["1058400","2744280","2757720","1421490","411600","67620","5880","210"],["13305600","36162720","38980920","22203720","7408170","1496880","180180","11880","330"],["179625600","508155120","580486500","358221600","133315875","31320135","4677750","430650","22275","495"],["2594592000","7599477600","9118806840","6012792500","2443104950","644969325","112807695","12977250","943800","39325","715"],["39956716800","120664383840","151068893976","105363334076","46041725730","13352874535","2640195558","357780423","32702670","1926925","66066","1001"],["653837184000","2028994531200","2636578788480","1930130322120","897087331140","281285754750","61411665315","9454855410","1023016995","76126050","3708705","106470","1365"],["11333177856000","36041021452800","48406025041920","36971030634720","18123020835920","6071736190520","1439516538460","245766380860","30338768460","2683540860","165785620","6790420","165620","1820"],["207484333056000","674648240716800","933333178821120","740154439752000","380137421518400","134858812888800","34294188568640","6381859984500","876819643700","88803214500","6544397860","341113500","11911900","249900","2380"]],"degrees":[-1,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"row_sums":["0","2","30","360","4200","50400","635040","8467200","119750400","1796256000","28540512000","479480601600","8499883392000","158664489984000","3112264995840000"],"nonn":true,"definitions":[{"position":0,"source":"oeis_catalog:A216118","source_kind":"oeis_catalog","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["0"],["1","1"]],"recurrence":{"denominator":{"coeffs":[["-1"],["1"]]},"terms":[{"coeff":{"coeffs":[["0","3"],["3","1"],["1","0"]]},"deriv_order":0,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":169731,"position":0,"claim":"real_rooted","from_index":1,"region_key":"(-inf, 0]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":1,"region":{"upper":{"closed":true,"value":"0"}}},"label":"Real-rooted","description":"Every row P_n, n >= 1, is zero or real-rooted with all roots in (-inf, 0]","region_exact":null,"edges":["e-b0244a114d676405"]},{"id":169732,"position":1,"claim":"one_signed_coefficients","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"alternating":false,"claim":"one_signed_coefficients","from_index":1},"label":"Coefficients of one sign","description":"Every nonzero row, n >= 1: the coefficients of P_n have one sign","region_exact":null,"edges":["e-b0244a114d676405"]},{"id":169733,"position":2,"claim":"hadamard_kernel","from_index":1,"region_key":null,"provenance":"derived_from_edges","claim_data":{"claim":"hadamard_kernel","from_index":1},"label":"Hadamard kernel (Malo)","description":"Rows P_n, n >= 1, are Hadamard kernels (preserve real-rootedness)","region_exact":null,"edges":["e-b0244a114d676405"]},{"id":169734,"position":3,"claim":"interlacing","from_index":2,"region_key":null,"provenance":"edges","claim_data":{"claim":"interlacing","from_index":2,"kind":"weak","orientation":"previous_interlaces_next"},"label":"Interlacing of consecutive rows","description":"Rows P_(m-1), P_m interlace (weak, previous interlaces next) for every m >= 2","region_exact":null,"edges":["e-b0244a114d676405"]}],"region_exact":{},"incoming_edges":[{"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","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":[]}],"used_by":[],"used_by_total":0,"links":[{"id":"l-33949a8fcb487fd8","kind":"fingerprint_related","node_a_id":"n-5b94261d6f603245","node_b_id":"n-a21fa1ea54a634a7","fingerprint":"f6-34eaf45657600f7a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-34eaf45657600f7a"},"other_id":"n-5b94261d6f603245","other":{"id":"n-5b94261d6f603245","name":"basis_map_A013610 of Triangle read by rows, s_3(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.","first_index":0,"fingerprint":"f6-34eaf45657600f7a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["3","3"],["18","27","9"]],"card_rows":[["1"],["3","3"],["18","27","9"],["162","297","162","27"]],"row_sums_preview":["1","6","54","648","9720","174960","3674160","88179840"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-8229dbdd6a4b5cd9","steps":["basis A013610"],"edges":["e-86f8b58a2aa07621"],"text":"A225470 \u2192 basis A013610"},"headline":"A225470 \u2192 basis A013610","chain":null}},{"id":"l-43952c2a93da08d8","kind":"fingerprint_related","node_a_id":"n-ab4410127fe29869","node_b_id":"n-a21fa1ea54a634a7","fingerprint":"f6-34eaf45657600f7a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-34eaf45657600f7a"},"other_id":"n-ab4410127fe29869","other":{"id":"n-ab4410127fe29869","name":"reversal of Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.","first_index":0,"fingerprint":"f6-34eaf45657600f7a","polytool_family":null,"degree_pattern":"irregular","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1"],["1","1"]],"card_rows":[["1"],["1"],["1","1"],["1","3","2"]],"row_sums_preview":["1","1","2","6","24","120","720","5040"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-b9fbfb57d9269e2f","steps":["reverse"],"edges":["e-c7203dc00a9aba0b"],"text":"A132393 \u2192 reverse"},"headline":"A132393 \u2192 reverse","chain":null}},{"id":"l-59837b3d10707b6a","kind":"fingerprint_related","node_a_id":"n-a21fa1ea54a634a7","node_b_id":"n-46edc3c082e19c85","fingerprint":"f6-34eaf45657600f7a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-34eaf45657600f7a"},"other_id":"n-46edc3c082e19c85","other":{"id":"n-46edc3c082e19c85","name":"Triangular array read by rows: T(n,k) = number of fixed points in the permutations of {1,2,...,n} that have exactly k cycles; n>=1, 1<=k<=n.","first_index":0,"fingerprint":"f6-34eaf45657600f7a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["0","2"],["0","3","3"]],"card_rows":[["1"],["0","2"],["0","3","3"],["0","8","12","4"]],"row_sums_preview":["1","2","6","24","120","720","5040","40320"],"regions":["(-inf, 1/2]"],"region_exact":{},"oeis_ids":["A180013"],"derivation":{"root":"n-4f6e747b794bcc2e","steps":["x \u21a6 2x\u22121"],"edges":["e-3176274160cea224"],"text":"A028338 \u2192 x \u21a6 2x\u22121"},"headline":"A180013: Triangular array read by rows: T(n,k) = number of fixed points in the permutations of {1,2,...,n} that have exactly k cycles; n>=1, 1<=k<=n.","chain":"A028338 \u2192 x \u21a6 2x\u22121"}},{"id":"l-75ddabfe1be8a510","kind":"fingerprint_related","node_a_id":"n-c2e61578a2a87aa5","node_b_id":"n-a21fa1ea54a634a7","fingerprint":"f6-34eaf45657600f7a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-34eaf45657600f7a"},"other_id":"n-c2e61578a2a87aa5","other":{"id":"n-c2e61578a2a87aa5","name":"basis_map_A013609 of Triangle of coefficients in expansion of (x+1)*(x+3)*...*(x + 2n - 1) in rising powers of x.","first_index":0,"fingerprint":"f6-34eaf45657600f7a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["2","2"],["8","12","4"]],"card_rows":[["1"],["2","2"],["8","12","4"],["48","88","48","8"]],"row_sums_preview":["1","4","24","192","1920","23040","322560","5160960"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-4f6e747b794bcc2e","steps":["basis A013609"],"edges":["e-03658933e548767a"],"text":"A028338 \u2192 basis A013609"},"headline":"A028338 \u2192 basis A013609","chain":null}},{"id":"l-7db6e9b90a917558","kind":"fingerprint_related","node_a_id":"n-4d60c3dd8627c01d","node_b_id":"n-a21fa1ea54a634a7","fingerprint":"f6-34eaf45657600f7a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-34eaf45657600f7a"},"other_id":"n-4d60c3dd8627c01d","other":{"id":"n-4d60c3dd8627c01d","name":"Triangle read by rows: T(n,k) = |s(n,n+1-k)|, where s(n,k) are the signed Stirling numbers of the first kind A008276 (1 <= k <= n; in other words, the unsigned Stirling numbers of the first kind in reverse order).","first_index":0,"fingerprint":"f6-34eaf45657600f7a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","1"],["1","3","2"]],"card_rows":[["1"],["1","1"],["1","3","2"],["1","6","11","6"]],"row_sums_preview":["1","2","6","24","120","720","5040","40320"],"regions":["(-inf, 0)","(-inf, 0]"],"region_exact":{},"oeis_ids":["A094638"],"derivation":{"root":"n-4d60c3dd8627c01d","steps":[],"edges":[],"text":"A094638"},"headline":"A094638: Triangle read by rows: T(n,k) = |s(n,n+1-k)|, where s(n,k) are the signed Stirling numbers of the first kind A008276 (1 <= k <= n; in other words, the unsigned Stirling numbers of the first kind in reverse order).","chain":null}},{"id":"l-853c380623d0686b","kind":"fingerprint_related","node_a_id":"n-f4e656360cd62235","node_b_id":"n-a21fa1ea54a634a7","fingerprint":"f6-34eaf45657600f7a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-34eaf45657600f7a"},"other_id":"n-f4e656360cd62235","other":{"id":"n-f4e656360cd62235","name":"basis_map_A013611 of Triangle read by rows, s_4(n, k) where s_m(n, k) are the Stirling-Frobenius cycle numbers of order m; n >= 0, k >= 0.","first_index":0,"fingerprint":"f6-34eaf45657600f7a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["4","4"],["32","48","16"]],"card_rows":[["1"],["4","4"],["32","48","16"],["384","704","384","64"]],"row_sums_preview":["1","8","96","1536","30720","737280","20643840","660602880"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":[],"derivation":{"root":"n-9c5c208df8605708","steps":["basis A013611"],"edges":["e-8a4919ac2dd1c88f"],"text":"A225471 \u2192 basis A013611"},"headline":"A225471 \u2192 basis A013611","chain":null}},{"id":"l-cdbc708a23dd25c3","kind":"fingerprint_related","node_a_id":"n-efe23a76953cf69d","node_b_id":"n-a21fa1ea54a634a7","fingerprint":"f6-34eaf45657600f7a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-34eaf45657600f7a"},"other_id":"n-efe23a76953cf69d","other":{"id":"n-efe23a76953cf69d","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.","first_index":0,"fingerprint":"f6-34eaf45657600f7a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["1","1"],["2","3","1"]],"card_rows":[["1"],["1","1"],["2","3","1"],["6","11","6","1"]],"row_sums_preview":["1","2","6","24","120","720","5040","40320"],"regions":["(-inf, 0)","(-inf, 0]","(-inf, -1]"],"region_exact":{},"oeis_ids":["A130534"],"derivation":{"root":"n-efe23a76953cf69d","steps":[],"edges":[],"text":"A130534"},"headline":"A130534: 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.","chain":null}},{"id":"l-f79ac9fc922c17d3","kind":"fingerprint_related","node_a_id":"n-b9fbfb57d9269e2f","node_b_id":"n-a21fa1ea54a634a7","fingerprint":"f6-34eaf45657600f7a","oeis_id":null,"in_catalog":null,"target":null,"row_alignment":null,"rows_matched":null,"oeis_nonn":null,"details":{"fingerprint":"f6-34eaf45657600f7a"},"other_id":"n-b9fbfb57d9269e2f","other":{"id":"n-b9fbfb57d9269e2f","name":"Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.","first_index":0,"fingerprint":"f6-34eaf45657600f7a","polytool_family":null,"degree_pattern":"affine","palindromic":0,"gamma_positive":0,"nonnegative":1,"rr_by_factors":1,"preview_rows":[["1"],["0","1"],["0","1","1"]],"card_rows":[["1"],["0","1"],["0","1","1"],["0","2","3","1"]],"row_sums_preview":["1","1","2","6","24","120","720","5040"],"regions":["(-inf, 0]"],"region_exact":{},"oeis_ids":["A132393"],"derivation":{"root":"n-b9fbfb57d9269e2f","steps":[],"edges":[],"text":"A132393"},"headline":"A132393: Triangle of unsigned Stirling numbers of the first kind (see A048994), read by rows, T(n,k) for 0 <= k <= n.","chain":null}},{"id":"l-d3b23381816f13d3","kind":"oeis_match","node_a_id":"n-a21fa1ea54a634a7","node_b_id":null,"fingerprint":null,"oeis_id":"A216119","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":1,"oeis_term_position":2,"target":"row_sums","terms_compared":15},"in_catalog":false,"oeis_id":"A216119","oeis_nonn":true},"other_id":null,"other":null}],"oeis":[{"oeis_id":"A216118","relation":"identification"},{"oeis_id":"A216119","relation":"match_row_sums"}],"bfiles":[],"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:         0;\n  1:         1,         1;\n  2:        10,        15,         5;\n  3:        90,       165,        90,        15;\n  4:       840,      1750,      1225,       350,        35;\n  5:      8400,     19180,     15750,      5950,      1050,       70;\n  6:     90720,    222264,    204624,     92610,     22050,     2646,     126;\n  7:   1058400,   2744280,   2757720,   1421490,    411600,    67620,    5880,    210;\n  8:  13305600,  36162720,  38980920,  22203720,   7408170,  1496880,  180180,  11880,   330;\n  9: 179625600, 508155120, 580486500, 358221600, 133315875, 31320135, 4677750, 430650, 22275, 495;\n","recurrence_sentence":"The row polynomials satisfy (n-1)*P(n,x) = (n^2+3*n+(n+3)*x)*P(n-1,x) for n >= 2, with P(0,x) = 0, P(1,x) = 1 + x.","mathematica":"P[0, x_] := 0;\nP[1, x_] := 1 + x;\nP[n_, x_] := P[n, x] = Expand[Together[((n^2 + 3*n + (n + 3)*x)*P[n - 1, x])/(n - 1)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 0; v[2] = 1 + x; for(n = 2, N, my(i = n + 1); v[i] = ((n^2 + 3*n + (n + 3)*x)*v[i - 1])/(n - 1)); 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-a21fa1ea54a634a7","steps":[],"edges":[],"text":"A216118"},"headline":"A216118: 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).","chain":null}