{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,22]],"date-time":"2026-04-22T05:45:48Z","timestamp":1776836748758,"version":"3.51.2"},"reference-count":29,"publisher":"American Mathematical Society (AMS)","issue":"341","license":[{"start":{"date-parts":[[2024,1,31]],"date-time":"2024-01-31T00:00:00Z","timestamp":1706659200000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    In this paper, we give an iterated approach to concern with the positivity of\n                    <disp-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"det left-parenthesis p left-parenthesis n minus i plus j right-parenthesis right-parenthesis Subscript 1 less-than-or-equal-to i comma j less-than-or-equal-to k Baseline comma\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo movablelimits=\"true\" form=\"prefix\">det<\/mml:mo>\n                            <mml:mtext>\u00a0<\/mml:mtext>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>p<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mi>i<\/mml:mi>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mi>j<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:msub>\n                              <mml:mo stretchy=\"false\">)<\/mml:mo>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>1<\/mml:mn>\n                                <mml:mo>\n                                  \u2264\n                                  \n                                <\/mml:mo>\n                                <mml:mi>i<\/mml:mi>\n                                <mml:mo>,<\/mml:mo>\n                                <mml:mi>j<\/mml:mi>\n                                <mml:mo>\n                                  \u2264\n                                  \n                                <\/mml:mo>\n                                <mml:mi>k<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo>,<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\begin{equation*} \\det \\ (p(n-i+j))_{1\\leq i,j\\leq k}, \\end{equation*}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/disp-formula>\n                    where\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"p left-parenthesis n right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>p<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">p(n)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    is the partition function. We first apply a general method to prove that for given\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"k 1 comma k 2 comma m 1 comma m 2\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:msub>\n                              <mml:mi>k<\/mml:mi>\n                              <mml:mn>1<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>k<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>m<\/mml:mi>\n                              <mml:mn>1<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>m<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msub>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">k_1,k_2,m_1,m_2<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    , one can find a threshold\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"upper N left-parenthesis k 1 comma k 2 comma m 1 comma m 2 right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>N<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>k<\/mml:mi>\n                              <mml:mn>1<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>k<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>m<\/mml:mi>\n                              <mml:mn>1<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>m<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">N(k_1,k_2,m_1,m_2)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    such that for\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"n greater-than upper N left-parenthesis k 1 comma k 2 comma m 1 comma m 2 right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mi>N<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>k<\/mml:mi>\n                              <mml:mn>1<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>k<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>m<\/mml:mi>\n                              <mml:mn>1<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:mo>,<\/mml:mo>\n                            <mml:msub>\n                              <mml:mi>m<\/mml:mi>\n                              <mml:mn>2<\/mml:mn>\n                            <\/mml:msub>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">n&gt;N(k_1,k_2,m_1,m_2)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ,\n                    <disp-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"Start 3 By 3 Determinant 1st Row 1st Column p left-parenthesis n minus k 1 plus m 1 right-parenthesis 2nd Column a m p semicolon p left-parenthesis n plus m 1 right-parenthesis 3rd Column a m p semicolon p left-parenthesis n plus m 1 plus m 2 right-parenthesis 2nd Row 1st Column p left-parenthesis n minus k 1 right-parenthesis 2nd Column a m p semicolon p left-parenthesis n right-parenthesis 3rd Column a m p semicolon p left-parenthesis n plus m 2 right-parenthesis 3rd Row 1st Column p left-parenthesis n minus k 1 minus k 2 right-parenthesis 2nd Column a m p semicolon p left-parenthesis n minus k 2 right-parenthesis 3rd Column a m p semicolon p left-parenthesis n minus k 2 plus m 2 right-parenthesis EndDeterminant greater-than 0 period\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mrow>\n                              <mml:mo>|<\/mml:mo>\n                              <mml:mtable rowspacing=\"4pt\" columnspacing=\"1em\">\n                                <mml:mtr>\n                                  <mml:mtd>\n                                    <mml:mi>p<\/mml:mi>\n                                    <mml:mo stretchy=\"false\">(<\/mml:mo>\n                                    <mml:mi>n<\/mml:mi>\n                                    <mml:mo>\n                                      \u2212\n                                      \n                                    <\/mml:mo>\n                                    <mml:msub>\n                                      <mml:mi>k<\/mml:mi>\n                                      <mml:mn>1<\/mml:mn>\n                                    <\/mml:msub>\n                                    <mml:mo>+<\/mml:mo>\n                                    <mml:msub>\n                                      <mml:mi>m<\/mml:mi>\n                                      <mml:mn>1<\/mml:mn>\n                                    <\/mml:msub>\n                                    <mml:mo stretchy=\"false\">)<\/mml:mo>\n                                  <\/mml:mtd>\n                                  <mml:mtd>\n                                    <mml:mi>a<\/mml:mi>\n                                    <mml:mi>m<\/mml:mi>\n                                    <mml:mi>p<\/mml:mi>\n                                    <mml:mo>;<\/mml:mo>\n                                    <mml:mi>p<\/mml:mi>\n                                    <mml:mo stretchy=\"false\">(<\/mml:mo>\n                                    <mml:mi>n<\/mml:mi>\n                                    <mml:mo>+<\/mml:mo>\n                                    <mml:msub>\n                                      <mml:mi>m<\/mml:mi>\n                                      <mml:mn>1<\/mml:mn>\n                                    <\/mml:msub>\n                                    <mml:mo stretchy=\"false\">)<\/mml:mo>\n                                  <\/mml:mtd>\n                                  <mml:mtd>\n                                    <mml:mi>a<\/mml:mi>\n                                    <mml:mi>m<\/mml:mi>\n                                    <mml:mi>p<\/mml:mi>\n                                    <mml:mo>;<\/mml:mo>\n                                    <mml:mi>p<\/mml:mi>\n                                    <mml:mo stretchy=\"false\">(<\/mml:mo>\n                                    <mml:mi>n<\/mml:mi>\n                                    <mml:mo>+<\/mml:mo>\n                                    <mml:msub>\n                                      <mml:mi>m<\/mml:mi>\n                                      <mml:mn>1<\/mml:mn>\n                                    <\/mml:msub>\n                                    <mml:mo>+<\/mml:mo>\n                                    <mml:msub>\n                                      <mml:mi>m<\/mml:mi>\n                                      <mml:mn>2<\/mml:mn>\n                                    <\/mml:msub>\n                                    <mml:mo stretchy=\"false\">)<\/mml:mo>\n                                  <\/mml:mtd>\n                                <\/mml:mtr>\n                                <mml:mtr>\n                                  <mml:mtd>\n                                    <mml:mi>p<\/mml:mi>\n                                    <mml:mo stretchy=\"false\">(<\/mml:mo>\n                                    <mml:mi>n<\/mml:mi>\n                                    <mml:mo>\n                                      \u2212\n                                      \n                                    <\/mml:mo>\n                                    <mml:msub>\n                                      <mml:mi>k<\/mml:mi>\n                                      <mml:mn>1<\/mml:mn>\n                                    <\/mml:msub>\n                                    <mml:mo stretchy=\"false\">)<\/mml:mo>\n                                  <\/mml:mtd>\n                                  <mml:mtd>\n                                    <mml:mi>a<\/mml:mi>\n                                    <mml:mi>m<\/mml:mi>\n                                    <mml:mi>p<\/mml:mi>\n                                    <mml:mo>;<\/mml:mo>\n                                    <mml:mi>p<\/mml:mi>\n                                    <mml:mo stretchy=\"false\">(<\/mml:mo>\n                                    <mml:mi>n<\/mml:mi>\n                                    <mml:mo stretchy=\"false\">)<\/mml:mo>\n                                  <\/mml:mtd>\n                                  <mml:mtd>\n                                    <mml:mi>a<\/mml:mi>\n                                    <mml:mi>m<\/mml:mi>\n                                    <mml:mi>p<\/mml:mi>\n                                    <mml:mo>;<\/mml:mo>\n                                    <mml:mi>p<\/mml:mi>\n                                    <mml:mo stretchy=\"false\">(<\/mml:mo>\n                                    <mml:mi>n<\/mml:mi>\n                                    <mml:mo>+<\/mml:mo>\n                                    <mml:msub>\n                                      <mml:mi>m<\/mml:mi>\n                                      <mml:mn>2<\/mml:mn>\n                                    <\/mml:msub>\n                                    <mml:mo stretchy=\"false\">)<\/mml:mo>\n                                  <\/mml:mtd>\n                                <\/mml:mtr>\n                                <mml:mtr>\n                                  <mml:mtd>\n                                    <mml:mi>p<\/mml:mi>\n                                    <mml:mo stretchy=\"false\">(<\/mml:mo>\n                                    <mml:mi>n<\/mml:mi>\n                                    <mml:mo>\n                                      \u2212\n                                      \n                                    <\/mml:mo>\n                                    <mml:msub>\n                                      <mml:mi>k<\/mml:mi>\n                                      <mml:mn>1<\/mml:mn>\n                                    <\/mml:msub>\n                                    <mml:mo>\n                                      \u2212\n                                      \n                                    <\/mml:mo>\n                                    <mml:msub>\n                                      <mml:mi>k<\/mml:mi>\n                                      <mml:mn>2<\/mml:mn>\n                                    <\/mml:msub>\n                                    <mml:mo stretchy=\"false\">)<\/mml:mo>\n                                  <\/mml:mtd>\n                                  <mml:mtd>\n                                    <mml:mi>a<\/mml:mi>\n                                    <mml:mi>m<\/mml:mi>\n                                    <mml:mi>p<\/mml:mi>\n                                    <mml:mo>;<\/mml:mo>\n                                    <mml:mi>p<\/mml:mi>\n                                    <mml:mo stretchy=\"false\">(<\/mml:mo>\n                                    <mml:mi>n<\/mml:mi>\n                                    <mml:mo>\n                                      \u2212\n                                      \n                                    <\/mml:mo>\n                                    <mml:msub>\n                                      <mml:mi>k<\/mml:mi>\n                                      <mml:mn>2<\/mml:mn>\n                                    <\/mml:msub>\n                                    <mml:mo stretchy=\"false\">)<\/mml:mo>\n                                  <\/mml:mtd>\n                                  <mml:mtd>\n                                    <mml:mi>a<\/mml:mi>\n                                    <mml:mi>m<\/mml:mi>\n                                    <mml:mi>p<\/mml:mi>\n                                    <mml:mo>;<\/mml:mo>\n                                    <mml:mi>p<\/mml:mi>\n                                    <mml:mo stretchy=\"false\">(<\/mml:mo>\n                                    <mml:mi>n<\/mml:mi>\n                                    <mml:mo>\n                                      \u2212\n                                      \n                                    <\/mml:mo>\n                                    <mml:msub>\n                                      <mml:mi>k<\/mml:mi>\n                                      <mml:mn>2<\/mml:mn>\n                                    <\/mml:msub>\n                                    <mml:mo>+<\/mml:mo>\n                                    <mml:msub>\n                                      <mml:mi>m<\/mml:mi>\n                                      <mml:mn>2<\/mml:mn>\n                                    <\/mml:msub>\n                                    <mml:mo stretchy=\"false\">)<\/mml:mo>\n                                  <\/mml:mtd>\n                                <\/mml:mtr>\n                              <\/mml:mtable>\n                              <mml:mo>|<\/mml:mo>\n                            <\/mml:mrow>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mn>0.<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\begin{equation*} \\begin {vmatrix} p(n-k_1+m_1) &amp; p(n+m_1) &amp; p(n+m_1+m_2)\\\\ p(n-k_1) &amp; p(n) &amp; p(n+m_2)\\\\ p(n-k_1-k_2) &amp; p(n-k_2) &amp; p(n-k_2+m_2) \\end{vmatrix}&gt;0. \\end{equation*}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/disp-formula>\n                    Based on this result, we will prove that for\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"n greater-than-or-equal-to 656\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo>\n                              \u2265\n                              \n                            <\/mml:mo>\n                            <mml:mn>656<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">n\\geq 656<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    ,\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"det left-parenthesis p left-parenthesis n minus i plus j right-parenthesis right-parenthesis Subscript 1 less-than-or-equal-to i comma j less-than-or-equal-to 4 Baseline greater-than 0\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo movablelimits=\"true\" form=\"prefix\">det<\/mml:mo>\n                            <mml:mtext>\u00a0<\/mml:mtext>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>p<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mi>i<\/mml:mi>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mi>j<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:msub>\n                              <mml:mo stretchy=\"false\">)<\/mml:mo>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>1<\/mml:mn>\n                                <mml:mo>\n                                  \u2264\n                                  \n                                <\/mml:mo>\n                                <mml:mi>i<\/mml:mi>\n                                <mml:mo>,<\/mml:mo>\n                                <mml:mi>j<\/mml:mi>\n                                <mml:mo>\n                                  \u2264\n                                  \n                                <\/mml:mo>\n                                <mml:mn>4<\/mml:mn>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                            <mml:mo>&gt;<\/mml:mo>\n                            <mml:mn>0<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\det \\ (p(n-i+j))_{1\\leq i,j\\leq 4}&gt;0<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . Employing the same technique, we will show that determinants\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"left-parenthesis p overbar left-parenthesis n minus i plus j right-parenthesis right-parenthesis Subscript 1 less-than-or-equal-to i comma j less-than-or-equal-to k\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mover>\n                                  <mml:mi>p<\/mml:mi>\n                                  <mml:mo stretchy=\"false\">\n                                    \u00af\n                                    \n                                  <\/mml:mo>\n                                <\/mml:mover>\n                              <\/mml:mrow>\n                            <\/mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mi>i<\/mml:mi>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mi>j<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:msub>\n                              <mml:mo stretchy=\"false\">)<\/mml:mo>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>1<\/mml:mn>\n                                <mml:mo>\n                                  \u2264\n                                  \n                                <\/mml:mo>\n                                <mml:mi>i<\/mml:mi>\n                                <mml:mo>,<\/mml:mo>\n                                <mml:mi>j<\/mml:mi>\n                                <mml:mo>\n                                  \u2264\n                                  \n                                <\/mml:mo>\n                                <mml:mi>k<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">({\\bar p}(n-i+j))_{1\\leq i,j\\leq k}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    are positive for\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"k equals 3 and 4\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi>k<\/mml:mi>\n                            <mml:mo>=<\/mml:mo>\n                            <mml:mn>3<\/mml:mn>\n                            <mml:mtext>\u00a0and\u00a0<\/mml:mtext>\n                            <mml:mn>4<\/mml:mn>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">k=3 \\text { and } 4<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    for overpartition\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"ModifyingAbove p With bar left-parenthesis n right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mover>\n                                  <mml:mi>p<\/mml:mi>\n                                  <mml:mo stretchy=\"false\">\n                                    \u00af\n                                    \n                                  <\/mml:mo>\n                                <\/mml:mover>\n                              <\/mml:mrow>\n                            <\/mml:mrow>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">{\\bar p}(n)<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    . Furthermore, we will give an outline of how to prove the positivity of\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"det left-parenthesis p left-parenthesis n minus i plus j right-parenthesis right-parenthesis Subscript 1 less-than-or-equal-to i comma j less-than-or-equal-to k Baseline\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mo movablelimits=\"true\" form=\"prefix\">det<\/mml:mo>\n                            <mml:mtext>\u00a0<\/mml:mtext>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>p<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi>n<\/mml:mi>\n                            <mml:mo>\n                              \u2212\n                              \n                            <\/mml:mo>\n                            <mml:mi>i<\/mml:mi>\n                            <mml:mo>+<\/mml:mo>\n                            <mml:mi>j<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                            <mml:msub>\n                              <mml:mo stretchy=\"false\">)<\/mml:mo>\n                              <mml:mrow class=\"MJX-TeXAtom-ORD\">\n                                <mml:mn>1<\/mml:mn>\n                                <mml:mo>\n                                  \u2264\n                                  \n                                <\/mml:mo>\n                                <mml:mi>i<\/mml:mi>\n                                <mml:mo>,<\/mml:mo>\n                                <mml:mi>j<\/mml:mi>\n                                <mml:mo>\n                                  \u2264\n                                  \n                                <\/mml:mo>\n                                <mml:mi>k<\/mml:mi>\n                              <\/mml:mrow>\n                            <\/mml:msub>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\det \\ (p(n-i+j))_{1\\leq i,j\\leq k}<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    for general\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"k\">\n                        <mml:semantics>\n                          <mml:mi>k<\/mml:mi>\n                          <mml:annotation encoding=\"application\/x-tex\">k<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    .\n                  <\/p>","DOI":"10.1090\/mcom\/3810","type":"journal-article","created":{"date-parts":[[2022,12,14]],"date-time":"2022-12-14T11:42:15Z","timestamp":1671018135000},"page":"1383-1402","source":"Crossref","is-referenced-by-count":7,"title":["Positivity of the determinants of the partition function and the overpartition function"],"prefix":"10.1090","volume":"92","author":[{"given":"Larry","family":"Wang","sequence":"first","affiliation":[]},{"given":"Neil","family":"Yang","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2023,1,31]]},"reference":[{"key":"1","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511617041","volume-title":"Irresistible integrals","author":"Boros, George","year":"2004","ISBN":"https:\/\/id.crossref.org\/isbn\/0521796369"},{"issue":"3","key":"2","doi-asserted-by":"publisher","first-page":"2143","DOI":"10.1090\/tran\/7707","article-title":"Higher order Tur\u00e1n inequalities for the partition function","volume":"372","author":"Chen, William Y. 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