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D preserves real-rootedness by Rolle's theorem and p + c p' is real-rooted for every real c by Rolle's theorem applied to e^(x/c) p; Polya--Schur 1914); when the coefficients of phi have one sign, coefficients of one sign stay of one sign; applied to phi(n, .) for each n","lean_route":null,"lean_declarations":["RealRooted.splits_add_C_mul_derivative_all","RealRooted.splits_C_mul_add_derivative","Polynomial.Splits.derivative"],"certificate_hash":"e5aea4c1c5d3ae625233b7d9b016be537eab28c0f39445e02dc5af088f8a91a4","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-5e3f8ad0b0a35d3e","role":"sequence","name":"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:    2,      1;\n  3:    6,      5;\n  4:   19,     28,      5;\n  5:   59,    180,     61;\n  6:  180,   1137,    662,     61;\n  7:  544,   6765,   7266,   1385;\n  8: 1637,  38176,  75354,  24568,  1385;\n  9: 4917, 207248, 719994, 408360, 50521;\n","recurrence_sentence":"The row polynomials satisfy (n-4+4*x)*P(n,x) = -(3*n-9+(2*n^2-16*n+42)*x+(n^3-11*n^2+46*n-82)*x^2+(6*n^2-42*n+68)*x^3+(8*n-16)*x^4)*P(n-2,x) - ((2*n-6)*x+(2*n^2-16*n+38)*x^2-(2*n^2-26*n+76)*x^3-(12*n-60)*x^4-16*x^5)*d/dx P(n-2,x) + (4*n-15+(2*n^2-16*n+48)*x+(10*n-42)*x^2+8*x^3)*P(n-1,x) + ((2*n-8)*x-(2*n-16)*x^2-8*x^3)*d/dx P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 1.","mathematica":"P[0, x_] := 1;\nP[1, x_] := 1;\nP[n_, x_] := P[n, x] = Expand[Together[(-(3*n - 9 + (2*n^2 - 16*n + 42)*x + (n^3 - 11*n^2 + 46*n - 82)*x^2 + (6*n^2 - 42*n + 68)*x^3 + (8*n - 16)*x^4)*P[n - 2, x] - ((2*n - 6)*x + (2*n^2 - 16*n + 38)*x^2 - (2*n^2 - 26*n + 76)*x^3 - (12*n - 60)*x^4 - 16*x^5)*D[P[n - 2, x], x] + (4*n - 15 + (2*n^2 - 16*n + 48)*x + (10*n - 42)*x^2 + 8*x^3)*P[n - 1, x] + ((2*n - 8)*x - (2*n - 16)*x^2 - 8*x^3)*D[P[n - 1, x], x])/(n - 4 + 4*x)]];\nTable[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten","pari":"P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 1; for(n = 2, N, my(i = n + 1); v[i] = (-(3*n - 9 + (2*n^2 - 16*n + 42)*x + (n^3 - 11*n^2 + 46*n - 82)*x^2 + (6*n^2 - 42*n + 68)*x^3 + (8*n - 16)*x^4)*v[i - 2] - ((2*n - 6)*x + (2*n^2 - 16*n + 38)*x^2 - (2*n^2 - 26*n + 76)*x^3 - (12*n - 60)*x^4 - 16*x^5)*deriv(v[i - 2], x) + (4*n - 15 + (2*n^2 - 16*n + 48)*x + (10*n - 42)*x^2 + 8*x^3)*v[i - 1] + ((2*n - 8)*x - (2*n - 16)*x^2 - 8*x^3)*deriv(v[i - 1], x))/(n - 4 + 4*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)","D+1"],"edges":["e-4c46966ead0b055f","e-0b45882766291d5a"],"text":"type_b_eulerian \u2192 \u03b3 (scale 1/4) \u2192 D+1"},"headline":"type_b_eulerian \u2192 \u03b3 (scale 1/4) \u2192 D+1","chain":null}