#Pólya frequency sequences
Pólya frequency sequences give a coefficient-level way to recognize real-rooted polynomials with non-negative coefficients. The basic idea is to put the coefficients into a Toeplitz matrix and ask for total non-negativity. This puts real-rootedness in the same framework as planar networks, interlacing polynomials, stable polynomials, and Hadamard products.
#Totally non-negative matrices
A matrix \(M=(m_{ij})\) is totally non-negative (TNN) if every minor is non-negative. It is totally positive (TP) if every minor is strictly positive.
TNN matrices arise naturally from path-counting in directed graphs via the Lindström–Gessel–Viennot lemma (LGV). Planar networks with compatible source and sink orderings yield TNN matrices automatically. As a standard example, Pascal’s triangle \(M_{n,j}=\binom{n}{j}\) is TNN by the LGV lemma applied to a grid with West and North steps. See [CFR25] for more on this viewpoint.
Example (TNN for a matrix via LGV).
The matrix \(M\) defined by \[M_{n,j} = \binom{d(n-j\ell)}{j}\] admits a lattice path interpretation. The entry \(M_{n,j}\) counts West–North paths from \(A_n=(dn,0)\) to \[B_j=((d\ell+1)j,j).\] Since sources lie on the \(x\)-axis with spacing \(d,\) and sinks lie on the line \(y=x/(d\ell+1),\) any non-identity matching forces a path crossing. By the LGV lemma, the matrix \((M_{n,j})\) is TNN.
The class of TNN matrices is closed under taking submatrices and under matrix multiplication. The latter follows from the Binet–Cauchy formula: each minor of a product is a sum of products of minors. These two closure properties are the engine behind many real-rootedness-preserving constructions below.
#Toeplitz matrices and PF sequences
Definition (Pólya frequency sequence).
A sequence \(a=(a_0,a_1,\dotsc)\) of non-negative real numbers is a Pólya frequency sequence (PF sequence) if the semi-infinite Toeplitz matrix \[T(a)_{i,j}=a_{i-j},\qquad a_k=0 \text{ for } k\lt{}0,\] is totally non-negative.
Thus a PF sequence is a sequence whose finite Toeplitz minors are all non-negative. If only minors of size at most \(r\) are required to be non-negative, the sequence is called \(\mathrm{PF}_r\). Thus \(\mathrm{PF}=\mathrm{PF}_\infty.\)
Proposition (Finite order two, see [Bre89, Kar68]).
A non-negative sequence is \(\mathrm{PF}_2\) if and only if it is log-concave and has no internal zeros.
Consequently every PF sequence is log-concave and unimodal. The converse is false: PF is a much stronger condition than log-concavity, because it asks for all Toeplitz minors, not only the adjacent \(2\times 2\) minors.
Theorem (Aissen–Schoenberg–Whitney, 1952).
A polynomial \[p(t)=\sum_k a_k t^k,\qquad a_k\geq 0,\] has only real non-positive roots if and only if its coefficient sequence \((a_0,a_1,\dotsc)\) is a Pólya frequency sequence. See [Kar68] for a full treatment.
This theorem is often the fastest way to move between real-rootedness and matrix total positivity. If a construction takes a TNN Toeplitz matrix to another TNN Toeplitz matrix, it automatically preserves real-rootedness of the corresponding polynomial.
Remark (PF sequences versus multiplier sequences).
The PF condition should be compared with, but not confused with, multiplier sequences. A PF sequence is a coefficient sequence: its ordinary generating function is tested by the Aissen–Schoenberg–Whitney theorem and its Toeplitz matrix is TNN. A multiplier sequence is an operator: the diagonal map \(T_\gamma(x^k)=\gamma_kx^k\) is tested by its exponential generating function, or equivalently by the Borcea–Brändén transcendental symbol. The common background is the Laguerre–Pólya/Pólya–Schur theory of real zeros.
Proposition (Closure under convolution).
If \(a=(a_n)_{n\geq 0}\) and \(b=(b_n)_{n\geq 0}\) are PF sequences, then their convolution \[(a*b)_n=\sum_{i+j=n} a_i b_j\] is again a PF sequence.
Proof
The Toeplitz matrix of the convolution is the product of Toeplitz matrices: \[T(a*b)=T(a)T(b).\] Since products of TNN matrices are TNN, \(T(a*b)\) is TNN.
For finite sequences, this is the coefficient-sequence version of the fact that the product of two polynomials with only real non-positive zeros again has only real non-positive zeros.
Theorem (Wang–Yeh PF preserver, [WY05, Cor. 2]).
Let \(x_0,x_1,\dotsc,x_{n-1}\) be a finite PF sequence, and set \(x_{-1}=x_n=0.\) For real \(a,b,c,d,\) define \[y_k=(a+c(k-1))x_{k-1}+(b+dk)x_k, \qquad 0\leq k\leq n.\] If \(ad\geq bc\) and every \(y_k\) is non-negative, then \(y_0,y_1,\dotsc,y_n\) is a PF sequence.
This gives a direct row-recursion test. Suppose a non-negative triangular array, with \(A(0,0)=1,\) satisfies \[A(n,k)=(rn+sk+t)A(n-1,k-1)+(an+bk+c)A(n-1,k).\] If \[rb\geq as \qquad\text{and}\qquad (r+s+t)b\geq(a+c)s,\] then every row \((A(n,0),\dotsc,A(n,n))\) is PF [WY05, Cor. 3]. This single criterion covers the binomial, Stirling, and Eulerian triangles, among many others.
#Edrei–Thoma theorem
The infinite-sequence version has a precise analytic form.
Theorem (Edrei–Thoma theorem, see [Kar68]).
A non-negative sequence \((a_n)_{n\geq 0}\) is a Pólya frequency sequence if and only if its ordinary generating function has the form \[\sum_{n\geq 0} a_n x^n = Cx^m e^{\gamma x} \prod_i \frac{1+\alpha_i x}{1-\beta_i x},\] where \(C,\gamma,\alpha_i,\beta_i\geq 0,\) \(m\geq 0,\) and \(\sum_i(\alpha_i+\beta_i)\lt{}\infty.\)
The polynomial case is the finite part of this classification: a polynomial with non-negative coefficients has coefficient sequence PF exactly when its zeros are real and non-positive.
#Variation-diminishing property
Total positivity was originally developed partly to control sign changes. For a finite real vector \(v,\) let \(\operatorname{var}(v)\) be the number of sign changes after deleting zero entries.
Theorem (Variation-diminishing property, see [Kar68, FJ11]).
If \(M\) is totally non-negative, then \[\operatorname{var}(Mv)\leq \operatorname{var}(v)\] for every real vector \(v\) for which the product is defined.
For a PF sequence, this says that convolution with the sequence cannot create new sign changes. This is another way to understand why PF sequences are rigid: the Toeplitz matrix controls not only the roots of one generating polynomial, but also the sign pattern of every sequence it acts on.
#Veronese sections
Theorem (Veronese sections preserve real-rootedness).
Let \(r\geq 1\) and \(0\leq i\lt{}r.\) Define the Veronese section \[\mathcal{S}^r_i\left(\sum_{m\geq 0} a_m t^m\right) = \sum_{m\geq 0} a_{i+rm} t^m.\] If \(f(t)\) is real-rooted with non-negative coefficients, then \(\mathcal{S}^r_i(f)\) is also real-rooted with non-negative coefficients.
Proof
The roots of \(f\) are necessarily non-positive. Hence, by Aissen–Schoenberg–Whitney, the coefficient sequence \((a_0,a_1,\dotsc)\) is PF, so its Toeplitz matrix \[T_{p,q}=a_{p-q}\] is TNN. The Toeplitz matrix for the subsequence \((a_i,a_{i+r},a_{i+2r},\dotsc)\) is the submatrix \[T_{i+rp,rq}=a_{i+r(p-q)}.\] It is therefore TNN as well. Applying Aissen–Schoenberg–Whitney again shows that \(\mathcal{S}^r_i(f)\) is real-rooted.
C. A. Athanasiadis and D. G. Wagner prove a stronger fully interlacing version of this statement [AW24, Thm. 5.3 and Rem. 5.4]. In particular, if \(f\) has non-negative coefficients and only real roots, then \[\bigl( \mathcal{S}^r_0(f), \mathcal{S}^r_1(f), \dotsc, \mathcal{S}^r_{r-1}(f) \bigr)\] is fully interlacing. Thus Veronese sections produce an interlacing family of polynomials, not only separate real-rooted polynomials. The earlier mutual-interlacing version appears in [Fis06, Thm. 7.65].
Example (Rows of Pascal’s triangle).
The polynomial \[\sum_{m=0}^{n}\binom{kn}{km}t^m\] is the Veronese section \(\mathcal{S}^k_0((1+t)^{kn}).\) Since \((1+t)^{kn}\) is real-rooted, the theorem above immediately gives that this polynomial is real-rooted.
#Path matrices and total positivity
Theorem (Characterization of TNN matrices, [Bre95, Thm. 3.1]).
A matrix is TNN if and only if it can be obtained from a fully compatible planar directed graph with non-negative edge weights.
The following result gives a structured family in which the same path model controls the array, its rows, and its columns.
Theorem (Cui–Zhu, [CZ23, Thm. 1.7]).
Fix \(t\geq 0\) and \(\ell\geq 1.\) Give each vertical step from height \(h-1\) to \(h\) a weight \(b_h\geq 0.\) For \(0\leq i\leq\ell,\) give every jump \((1,t+i)\) weight \(a_i\geq 0,\) where \[\sum_{i=0}^{\ell}a_i u^i =\prod_{j=1}^{\ell}(\alpha_j u+\beta_j), \qquad \alpha_j,\beta_j\geq 0.\] Let \(M_{n,k}\) be the total weight of paths from \((0,0)\) to \((k,n).\) Then \(M=(M_{n,k})_{n,k\geq 0}\) is TNN. If \(t\geq 1,\) the Toeplitz matrix of every row of \(M\) is TNN, so every row is PF.
If \(b_h=\gamma\geq 0\) for every \(h,\) the Toeplitz matrix of every column is also TNN, and \[M=\mathcal R\left( \frac{1}{1-\gamma z}, \frac{z^t\prod_{j=1}^{\ell}(\alpha_jz+\beta_j)}{1-\gamma z} \right).\]
The fixed factorization is essential: the jump-weight polynomial is the same at every height, hence has the same non-positive zeros throughout the path model. The first-kind Jacobi–Stirling triangle is a bidiagonal specialization of the related height-dependent path construction in [CZ23, Sec. 7.6].
Theorem (Characterization of TNN lower unitriangular matrices, [BL24]).
Let \(\Gamma_N\) be the directed graph on \(\{(i,j)\in\mathbb{N}^2 : j \leq i \leq N\}\) with horizontal edges \((i,j)\to(i,j+1)\) and vertical edges \((i+1,j)\to(i,j).\) Assign non-negative weights \(\lambda=(\lambda_{i,j})_{0\leq j\leq i\lt{}N}\) to the vertical edges (and weight \(1\) on horizontal edges), and let \(r_{n,k}(\Gamma_N,\lambda)\) be the total weight of paths from \((n,0)\) to \((k,k).\)
A lower unitriangular matrix \(R=(r_{n,k})_{n,k=0}^N\) is totally non-negative if and only if there exists such a weight array \(\lambda\) with \(R=R(\Gamma_N,\lambda).\) The forward direction follows from the LGV lemma; the converse is Whitney’s reduction theorem, see [FJ11].
Checking TNN. Total non-negativity can be checked efficiently via Neville elimination, a variant of Gaussian elimination using adjacent-row operations. A matrix is TNN if and only if Neville elimination completes with all multipliers non-negative and no entry becoming negative. This gives an \(O(n^2m)\) algorithm for an \(n\times m\) matrix, compared to the exponential cost of checking all minors directly.
Hadamard products do not preserve TNN in general, even for arbitrary TNN Toeplitz matrices. Two useful positive cases are the finite-support Schur–Pólya theorem and Wagner’s theorem for lower-triangular TNN Toeplitz matrices whose diagonal sequences are polynomial functions of the diagonal index [Wag92]. See [BL24] for applications related to chains in posets, and [FJ11] for a comprehensive treatment.
#Riordan arrays from PF data
The PF viewpoint also interacts well with Riordan arrays. If \[g(z)=\sum_{n\geq 0}g_nz^n,\qquad f(z)=\sum_{n\geq 0}f_nz^n,\] then the Riordan array \[\mathcal R(g,f)=(r_{n,k})_{n,k\geq 0}\] is defined by saying that the \(k\)th column has generating function \[\sum_{n\geq 0} r_{n,k}z^n=g(z)f(z)^k.\] Thus a Toeplitz matrix is the special case \(\mathcal R(g,z).\)
Theorem (Chen–Wang, [CW19, Thm. 2.1]).
If the coefficient sequences of \(g(z)\) and \(f(z)\) are PF, then the Riordan array \(\mathcal R(g,f)\) is totally non-negative.
This gives two complementary kinds of structure: \[\text{PF input} \quad\Longrightarrow\quad \text{TNN Riordan array},\] and, separately, \[\text{Riordan description} \quad\Longrightarrow\quad \text{row recurrences and row relations} \quad\Longrightarrow\quad \text{real-rootedness and interlacing}.\] The second implication is the one responsible for the root statements in the examples below. The PF condition explains the total-positivity of the coefficient array; the real-rootedness usually comes from a recurrence, explicit zero formula, or an interlacing comparison between row polynomials.
Example (Delannoy and coordination triangles).
H. Liang, Y. Pei, and Y. Wang [LPW23] use this perspective for coordination numbers of cubic lattices. The Delannoy triangle is \[\mathcal R\left(\frac{1}{1-z},\frac{z(1+z)}{1-z}\right).\] Since both coefficient sequences are PF, the triangle is TNN by the Riordan-array criterion; in fact it is lower-triangular strictly totally positive in the sense that every minor not forced to vanish by triangular support is positive.
If \(d_n(x)\) is the \(n\)th row polynomial of this triangle, then \[d_0(x)=1,\qquad d_1(x)=1+x,\qquad d_n(x)=(1+x)d_{n-1}(x)+x d_{n-2}(x)\quad(n\geq 2).\] The zeros of \(d_n(x)\) are explicit, real, and lie in \[\bigl(-3-2\sqrt{2},\,-3+2\sqrt{2}\bigr).\]
The cubic-lattice coordination triangle is \[\mathcal R\left(\frac{1+z}{1-z},\frac{z(1+z)}{1-z}\right).\] Its row polynomial \(c_n(x)\) satisfies \[c_n(x)=d_n(x)+d_{n-1}(x),\] and Liang–Pei–Wang prove that \(c_{n-1}(x)\) interlaces \(c_n(x).\) All zeros of \(c_n(x)\) are real and lie in the same Delannoy interval.
Example (A one-parameter Delannoy family).
For \(m\geq 1,\) set \[\widehat L^{(m)} = \mathcal R\left( \frac{1+(m-1)z}{1-z}, \frac{z(1+z)}{1-z} \right).\] Again the PF criterion gives total non-negativity. If \(\ell^{(m)}_n(x)\) denotes the \(n\)th row polynomial, then \[\ell^{(m)}_n(x)=d_n(x)+(m-1)d_{n-1}(x)\quad(n\geq 1).\] For \(m\geq 2,\) these polynomials have real simple zeros and form an interlacing chain [LPW23, Rem. 29].
Example (Type \(B\) Narayana rows).
The same PF/TNN method also explains the total positivity of the type \(B\) Narayana triangle \[N_B(n,k)=\binom{n}{k}^2 .\] Y. Wang and A. L. B. Yang [WY18] prove that the type \(A\) and type \(B\) Narayana triangles are TNN, and that the corresponding Narayana squares are strictly totally positive. The row polynomials \[\sum_{k=0}^{n}\binom{n}{k}^2x^k\] are the type \(B\) Narayana polynomials, hence are real-rooted. Liang–Pei–Wang point out that this triangle is also the coordinator triangle for root lattices of type \(A.\)
#Examples
Example (Ray through Pascal’s triangle).
The following is proved in [Yu09]. Let \(n\geq k\geq 0\) and \(b\gt{}a\gt{}0,\) with \(k\lt{}b.\) Then the polynomial \[\sum_{j\geq 0} \binom{n+ja}{k+jb} t^j\] is real-rooted. The proof uses an interpretation via lattice paths and solves [FT16, Problem 12].
Example (Delannoy numbers).
Let \(\Delta(a,b)\) be the Delannoy number counting lattice paths from \((0,0)\) to \((a,b)\) with east, north, and north-east steps. Equivalently, \[\Delta(a,b)=\sum_i \binom{a}{i}\binom{b}{i}2^i .\] The square array \((\Delta(a,b))_{a,b\geq 0}\) is A008288 when read by antidiagonals, and it forms a TNN matrix [Bre95, Cor. 5.15]. The finite antidiagonal row polynomials \[P_n(t)=\sum_{k=0}^n \Delta(n-k,k)t^k\] begin \[1,\qquad 1+t,\qquad 1+3t+t^2,\qquad 1+5t+5t^2+t^3,\dotsc\] and satisfy \[P_0(t)=1,\qquad P_1(t)=1+t,\qquad P_n(t)=(1+t)P_{n-1}(t)+tP_{n-2}(t) \quad (n\geq 2).\] These are the Delannoy row polynomials that occur as \(h^*\)-polynomials of regular snake posets.
Theorem (Brenti [Bre95]).
Let \(D\) be a locally finite digraph on \(\setN\times\setN\) with non-negative edge weights. Assume that \(D\) is planar and weakly \(y\)-invariant, meaning that the edges from \((m,k)\) are independent of \(m.\) Let \[M_{n,j}\coloneqq P_D\bigl((0,0),(n,j)\bigr)\] be the weighted path count from \((0,0)\) to \((n,j).\) Then \(M=(M_{n,j})\) is totally positive and every row is a Pólya frequency sequence. In particular, every row polynomial \(\sum_j M_{n,j}t^j\) has only real non-positive zeros.
#Schur functions and Toeplitz minors
There is also a direct symmetric-function viewpoint. Let \(h_0=1,\) let \(h_m=0\) for \(m\lt{}0,\) and form the Toeplitz matrix \[H=(h_{i-j})_{i,j\geq 0}.\] The skew Jacobi–Trudi identity says that the minors of \(H\) are skew Schur functions, up to the harmless reversal of row and column order. Thus, after specializing the variables to non-negative real numbers, every minor is non-negative. Equivalently, the specialized sequence \[h_0(\xvec), h_1(\xvec), h_2(\xvec),\dotsc\] is PF. The same conclusion follows from Edrei–Thoma, since \[\sum_{n\geq 0} h_n(\xvec)t^n = \prod_i \frac{1}{1-x_it}.\]
Theorem (Stanley, [Sta98]).
A polynomial \(Q(x)=1+a_1x+a_2x^2+\dotsb+a_nx^n\) has only negative real roots if and only if the product \[\prod_{i\geq 1} Q(x_i)\] is Schur-positive.
Z. Fu, Y. Peng, and Y. Zhang connect this kind of total positivity with upper homogeneous posets, or upho posets [FPZ24]. They prove that a formal power series \(f(x)\in 1+x\setZ_{\geq 0}[[x]]\) is the rank-generating function of a Schur-positive upho poset if and only if \(f(x)\) is totally positive. Here Schur positivity refers to the Ehrenborg quasisymmetric function of the poset. Their proof passes through a correspondence between colored upho posets and atomic left-cancellative monoids.
Bibliography
- [AW24]Christos A. Athanasiadis and David G. Wagner. Veronese sections and interlacing matrices of polynomials and formal power series. arXiv:2404.12989, 2024.
.bib
@article{AthanasiadisWagner2024x, author = {Christos A. Athanasiadis and David G. Wagner}, title = {Veronese sections and interlacing matrices of polynomials and formal power series}, journal = {arXiv e-prints}, year = {2024}, eprint = {2404.12989}, doi = {10.48550/arXiv.2404.12989} } - [BL24]Petter Brändén and Leonardo Saud Maia Leite. Totally nonnegative matrices, chain enumeration and zeros of polynomials. arXiv:2412.06595, 2024.
.bib
@article{BrandenLeite2024x, author = {Petter Brändén and Leonardo Saud Maia Leite}, title = {Totally nonnegative matrices, chain enumeration and zeros of polynomials}, journal = {arXiv e-prints}, Eprint = {2412.06595}, url = {https://arxiv.org/abs/2412.06595}, year = {2024} } - [Bre89]Francesco Brenti. Unimodal, log-concave and Pólya frequency sequences in combinatorics. Mem. Amer. Math. Soc., 81(413):viii+106, 1989.
.bib
@article{Brenti1989, AUTHOR = {Brenti, Francesco}, TITLE = {Unimodal, log-concave and {P}{\'{o}}lya frequency sequences in combinatorics}, JOURNAL = {Mem. Amer. Math. Soc.}, FJOURNAL = {Memoirs of the American Mathematical Society}, VOLUME = {81}, YEAR = {1989}, NUMBER = {413}, PAGES = {viii+106}, ISSN = {0065-9266,1947-6221}, MRCLASS = {05A15 (05A10 05A20)}, MRNUMBER = {963833}, MRREVIEWER = {Ira\ Gessel}, DOI = {10.1090/memo/0413}, URL = {https://doi.org/10.1090/memo/0413}, } - [Bre95]Francesco Brenti. Combinatorics and total positivity. Journal of Combinatorial Theory, Series A, 71(2):175–218, 1995.
.bib
@article{Brenti1995, author = {Brenti, Francesco}, title = {Combinatorics and total positivity}, year = {1995}, journal = {Journal of Combinatorial Theory, Series A}, volume = {71}, number = {2}, pages = {175--218}, publisher = {Elsevier BV}, doi = {10.1016/0097-3165(95)90000-4}, url = {http://dx.doi.org/10.1016/0097-3165(95)90000-4}, issn = {0097-3165} } - [CFR25]Xi Chen, Lang Fu and Jiajie Ruan. A unified planar network approach to total positivity of combinatorial matrices and real-rootedness of polynomials. arXiv:2512.08369, 2025.
.bib
@article{ChenFuRuan2025x, author = {Xi Chen and Lang Fu and Jiajie Ruan}, title = {A unified planar network approach to total positivity of combinatorial matrices and real-rootedness of polynomials}, year = {2025}, eprint = {2512.08369}, url = {https://arxiv.org/abs/2512.08369}, journal = {arXiv e-prints} } - [CW19]Xi Chen and Yi Wang. Notes on the total positivity of Riordan arrays. Linear Algebra and its Applications, 569:156–161, 2019.
.bib
@article{ChenWang2019Riordan, author = {Chen, Xi and Wang, Yi}, title = {Notes on the total positivity of {R}iordan arrays}, year = {2019}, journal = {Linear Algebra and its Applications}, volume = {569}, pages = {156--161}, doi = {10.1016/j.laa.2019.01.015} } - [CZ23]Yu-Jie Cui and Bao-Xuan Zhu. Total positivity from a kind of lattice paths. arXiv:2308.05167v1, 2023.
.bib
@article{CuiZhu2023x, author = {Yu-Jie Cui and Bao-Xuan Zhu}, title = {Total positivity from a kind of lattice paths}, year = {2023}, eprint = {2308.05167v1}, url = {https://arxiv.org/abs/2308.05167v1}, journal = {arXiv e-prints} } - [FJ11]Shaun M. Fallat and Charles R. Johnson. Totally nonnegative matrices. Princeton series in applied mathematics. Princeton University Press, 2011.
.bib
@book{FallatJohnson2011, author = {Fallat, Shaun M. and Johnson, Charles R.}, title = {Totally Nonnegative Matrices}, publisher = {Princeton University Press}, series = {Princeton Series in Applied Mathematics}, year = {2011}, doi = {10.1515/9781400839018} } - [Fis06]Steve Fisk. Polynomials, roots, and interlacing. arXiv:math/0612833, 2006.
.bib
@article{Fisk2006x, Author = {Steve Fisk}, Title = {Polynomials, roots, and interlacing}, Year = {2006}, Eprint = {arXiv:math/0612833}, journal = {arXiv e-prints} } - [FT16]Tamás Forgács and Khang Tran. Polynomials with rational generating functions and real zeros. Journal of Mathematical Analysis and Applications, 443(2):631–651, November 2016.
.bib
@article{ForgcsTran2016, title = {Polynomials with rational generating functions and real zeros}, volume = {443}, ISSN = {0022-247X}, url = {http://dx.doi.org/10.1016/j.jmaa.2016.05.041}, DOI = {10.1016/j.jmaa.2016.05.041}, number = {2}, journal = {Journal of Mathematical Analysis and Applications}, publisher = {Elsevier BV}, author = {Forgács, Tamás and Tran, Khang}, year = {2016}, month = nov, pages = {631--651} } - [FPZ24]Ziyao Fu, Yulin Peng and Yuchong Zhang. The monoid representation of upho posets and total positivity. arXiv:2411.04123, 2024.
.bib
@article{FuPengZhang2024x, author = {Ziyao Fu and Yulin Peng and Yuchong Zhang}, title = {The monoid representation of upho posets and total positivity}, year = {2024}, eprint = {2411.04123}, url = {https://arxiv.org/abs/2411.04123}, journal = {arXiv e-prints} } - [Kar68]Samuel Karlin. Total positivity, Vol. I. Stanford University Press, 1968.
.bib
@book{Karlin1968, author = {Karlin, Samuel}, title = {Total Positivity, {V}ol.~{I}}, publisher = {Stanford University Press}, year = {1968} } - [LPW23]Huyile Liang, Yanni Pei and Yi Wang. Analytic combinatorics of coordination numbers of cubic lattices. Advances in Applied Mathematics, 147:102518, 2023.
.bib
@article{LiangPeiWang2023Coordination, author = {Liang, Huyile and Pei, Yanni and Wang, Yi}, title = {Analytic combinatorics of coordination numbers of cubic lattices}, year = {2023}, journal = {Advances in Applied Mathematics}, volume = {147}, pages = {102518}, doi = {10.1016/j.aam.2023.102518} } - [Sta98]Richard P. Stanley. Graph colorings and related symmetric functions: Ideas and applications. Discrete Mathematics, 193(1):267–286, 1998.
.bib
@article{Stanley1998, title = {Graph colorings and related symmetric functions: ideas and applications}, journal = {Discrete Mathematics}, volume = {193}, number = {1}, pages = {267--286}, year = {1998}, issn = {0012-365X}, doi = {10.1016/S0012-365X(98)00146-0}, url2 = {http://www.sciencedirect.com/science/article/pii/S0012365X98001460}, author = {Richard P. Stanley} } - [Wag92]David G Wagner. Total positivity of Hadamard products. Journal of Mathematical Analysis and Applications, 163(2):459–483, January 1992.
.bib
@article{Wagner1992, doi = {10.1016/0022-247x(92)90261-b}, url2 = {https://doi.org/10.1016/0022-247x(92)90261-b}, year = {1992}, month = jan, publisher = {Elsevier {BV}}, volume = {163}, number = {2}, pages = {459--483}, author = {David G Wagner}, title = {Total positivity of {H}adamard products}, journal = {Journal of Mathematical Analysis and Applications} } - [WY18]Yi Wang and Arthur L. B. Yang. Total positivity of Narayana matrices. Discrete Mathematics, 341(5):1264–1269, 2018.
.bib
@article{WangYang2018NarayanaMatrices, author = {Wang, Yi and Yang, Arthur L. B.}, title = {Total positivity of {N}arayana matrices}, year = {2018}, journal = {Discrete Mathematics}, volume = {341}, number = {5}, pages = {1264--1269}, doi = {10.1016/j.disc.2018.01.012} } - [WY05]Yi Wang and Yeong-Nan Yeh. Polynomials with real zeros and Pólya frequency sequences. Journal of Combinatorial Theory, Series A, 109(1):63–74, 2005.
.bib
@article{WangYeh2005, author = {Yi Wang and Yeong-Nan Yeh}, title = {Polynomials with real zeros and {P}{\'o}lya frequency sequences}, year = {2005}, journal = {Journal of Combinatorial Theory, Series A}, volume = {109}, number = {1}, pages = {63--74}, doi = {10.1016/j.jcta.2004.07.008}, url = {https://doi.org/10.1016/j.jcta.2004.07.008}, eprint = {math/0611825} } - [Yu09]Yaming Yu. Confirming two conjectures of Su and Wang on binomial coefficients. Advances in Applied Mathematics, 43(4):317–322, October 2009.
.bib
@article{Yu2009, doi = {10.1016/j.aam.2008.12.004}, url2 = {https://doi.org/10.1016/j.aam.2008.12.004}, year = {2009}, month = oct, publisher = {Elsevier {BV}}, volume = {43}, number = {4}, pages = {317--322}, author = {Yaming Yu}, title = {Confirming two conjectures of {S}u and {W}ang on binomial coefficients}, journal = {Advances in Applied Mathematics} }