{"id":"n-ca0966673c5193af","revision":0,"first_index":0,"fingerprint":"f6-c597530243fc1bdb","name":"shift_x_plus_one of 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.","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"],["3","1"],["11","5"],["57","38","5"],["361","302","61"],["2763","2827","845","61"],["24611","29607","11421","1385"],["250737","347372","165678","30108","1385"],["2873041","4501564","2551326","610444","50521"],["36581523","63967093","42044902","12558738","1578727","50521"],["512343611","988962409","739436006","264685994","44488351","2702765"],["7828053417","16532652626","13858225819","5787668396","1224886319","114545698","2702765"],["129570724921","297191234346","276031441155","131781304220","33659982735","4261950426","199360981"],["2309644635483","5717183815999","5828199544491","3131754984615","938210069225","150729398853","10991602449","199360981"]],"degrees":[0,0,1,1,2,2,3,3,4,4,5,5,6,6,7],"row_sums":["1","1","4","16","100","724","6496","67024","795280","10586896","156781504","2552619136","45348735040","872695998784","18086913412096"],"nonn":true,"definitions":[{"position":0,"source":"derived:shift_x_plus_one","source_kind":"derived","equality_edge_id":null,"recurrence":{"first_index":0,"initial_polynomials":[["1"]],"recurrence":{"terms":[{"coeff":{"coeffs":[["-1","-1"],["2","1"]]},"deriv_order":0,"offset":1,"sign":"none"},{"coeff":{"coeffs":[["-4","-6","-2"]]},"deriv_order":1,"offset":1,"sign":"none"}]},"schema":"polytool.recurrence.v1"}}],"properties":[{"id":177519,"position":0,"claim":"real_rooted","from_index":0,"region_key":"(-inf, -1]","provenance":"edges","claim_data":{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}},"label":"Real-rooted","description":"Every row P_n, n >= 0, is zero or real-rooted with all roots in (-inf, -1]","region_exact":null,"edges":["e-c3ee495a04f89a6a"]},{"id":177520,"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-c3ee495a04f89a6a"]},{"id":177521,"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-c3ee495a04f89a6a"]}],"region_exact":{},"incoming_edges":[{"id":"e-c3ee495a04f89a6a","operation":{"index_shift":0,"operation":"preserver_image","operator":"{\"shape\":\"affine\",\"slope\":\"1\",\"intercept\":\"1\"}"},"operation_detail":"affine, shift 0","conclusion_id":"n-ca0966673c5193af","premise_count":1,"theorem":"a real affine substitution x -> a x + b (a != 0), the reversal x^d p(1/x) (d >= deg p) and a real Mobius map (c x + e)^d p((a x + b)/(c x + e)) (a e - b c != 0, d >= deg p) map real roots to real roots: a root r of p gives the root (r - b)/a, 1/r (and 0) or (e r - b)/(a - c r) (and -e/c); for an input row that is a polynomial the degree condition holds automatically","lean_route":null,"lean_declarations":["Polynomial.Splits.comp_of_degree_le_one"],"certificate_hash":"ac696a52c1dcfb219f0efe1043a91650a66c7fc762b8ac0d84310a41ae5a3799","verification_polytool":"replayed","verification_lean":"not_attempted","claims":[{"claim":"real_rooted","from_index":0,"region":{"upper":{"closed":true,"value":"-1"}}}],"premises":[{"position":0,"node_id":"n-5d7c3f4772a9d8bb","role":"sequence","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.","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":[],"used_by_total":0,"links":[],"oeis":[],"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:       1;\n  1:       1;\n  2:       3,       1;\n  3:      11,       5;\n  4:      57,      38,       5;\n  5:     361,     302,      61;\n  6:    2763,    2827,     845,     61;\n  7:   24611,   29607,   11421,   1385;\n  8:  250737,  347372,  165678,  30108,  1385;\n  9: 2873041, 4501564, 2551326, 610444, 50521;\n","recurrence_sentence":"The row polynomials satisfy P(n,x) = (2*n-1+(n-1)*x)*P(n-1,x) - (4+6*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[(2*n - 1 + (n - 1)*x)*P[n - 1, x] - (4 + 6*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] = (2*n - 1 + (n - 1)*x)*v[i - 1] - (4 + 6*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)","x \u21a6 x+1","x \u21a6 x+1"],"edges":["e-16125ab4d60fcfd7","e-42ac1b21fb8d0456","e-c3ee495a04f89a6a"],"text":"type_b_eulerian \u2192 \u03b3 (scale 1/4) \u2192 x \u21a6 x+1 \u2192 x \u21a6 x+1"},"headline":"type_b_eulerian \u2192 \u03b3 (scale 1/4) \u2192 x \u21a6 x+1 \u2192 x \u21a6 x+1","chain":null}