#Permutation statistics, words, and set partitions
This page collects real-rootedness results for statistics on permutations, words, and set partitions. Many examples are Sturm sequences, meaning that \(P_j\interl P_{j+1}\) for all \(j.\) Others use interlacing sequences, stable polynomials, or total-positivity methods. For the broader context of real-rootedness and unimodality, see unimodality and real-rootedness. For descent polynomials of linear extensions of posets, including generalized snake posets, see poset operations preserving real-rootedness.
The examples are grouped into Eulerian-type descent polynomials, other permutation statistics, set partitions, and Stirling permutations. The subsections are meant as a guide to the methods as much as to the objects: repeated themes are derivative recurrences, compatible or interlacing refinements, and multivariate stability.
#Eulerian-type descent polynomials
#Classical and multiset Eulerian polynomials
Example (Eulerian polynomials, descents in permutations).
The Eulerian polynomials are \[A_n(t) \coloneqq \sum_{\sigma \in \symS_n} t^{\des(\sigma)}.\] These satisfy \[A_n(t) = (1+t(n-1))A_{n-1}(t) + t(1-t) A'_{n-1}(t), \quad A_1(t)=1\] and this recurrence implies that \(A_j(t)\interl A_{j+1}(t)\) for all \(j.\) The coefficient \(n-1\) is large enough that this is not a direct application of Wagner’s lemma; the root positions must be used more carefully.
This result goes back to Frobenius [Fro10].
The proof proceeds as follows. Introduce \(\tilde{A}_n(x) \coloneqq x A_n(x).\) The recursion then becomes \[\tilde{A}_n(x) = x(1-x)\tilde{A}'_{n-1}(x) + nx \tilde{A}_{n-1}(x).\] Since \(\tilde{A}'_{n-1}\interl \tilde{A}_{n-1},\) the two summands have controlled roots. The term \(x(1-x)\tilde{A}'_{n-1}(x)\) has negative leading coefficient, while the sum \[x(1-x)\tilde{A}'_{n-1}(x) + nx \tilde{A}_{n-1}(x)\] has a positive leading coefficient due to \(n \geq 2\) in the recursion. Drawing the roots on the line gives the desired interlacing.
Example (Multi-Eulerian polynomials).
Let \(\symS_M\) be the permutations of some multiset \(1^{\mu_1},2^{\mu_2},\dotsc,n^{\mu_n}\) with \(N=|\mu|.\) Define \[A_M(x;q) \coloneqq \sum_{\pi \in \symS_M } x^{\des(\pi)}q^{\maj(\pi)}.\] These are the multi-Eulerian polynomials. Then (MacMahon, [Mac]) \[\frac{A_M(x;q)}{\prod_{j=0}^N (1-x q^j)} = \sum_{m \geq 0} \left(\prod_{j=1}^n \qbinom{\mu_j+m}{m}_q \right) x^m.\] The polynomials \(A_M(x;1)\) are then real-rooted, see [Sim84, Section 2]. Ma and Pan [MP23, Thm. 1.11] give a proof of this using interlacing polynomials. For stability in multivariate Eulerian versions, see the stable-polynomial page and [BHVW11, Eq. (4.7)].
D. Deligeorgaki, B. Han, and L. Solus study colored multiset Eulerian polynomials, which simultaneously generalize MacMahon’s multiset Eulerian polynomials and colored Eulerian polynomials [DHS26]. They characterize the symmetric cases and give sufficient conditions for self-interlacing; this implies real-rootedness, log-concavity, alternating increase, and bi-gamma-positivity. Their proofs use multivariate generalizations of MacMahon’s identity above.
X. Yan subsequently proved real-rootedness for all positive multiplicity and color vectors, without the self-interlacing restrictions above. Both the descent polynomial and its sentinel-ascent analogue are real-rooted, and increasing one multiplicity gives an interlacing in each family [Yan26, Thms. 1.4–1.5].
Example (Multiset Eulerian–Narayana polynomials).
Let \[M=\{1^{p_1},\dotsc,m^{p_m}\}, \qquad p_i\geq1, \qquad N=|M|.\] A weakly increasing tree on \(M\) is a plane tree on \(M\uplus\{0\},\) rooted at \(0,\) whose labels increase weakly along every root-to-leaf path and from left to right among siblings. If \(\mathcal{T}_M\) is the set of these trees, define the multiset Eulerian–Narayana polynomial by \[A_M(t)\coloneqq \sum_{T\in\mathcal{T}_M}t^{\operatorname{leaf}(T)}, \qquad P_M(t)\coloneqq A_M(t)/t.\] The extreme cases \(M=\{1^N\}\) and \(M=\{1,\dotsc,N\}\) give the Narayana and Eulerian polynomials, respectively.
P. B. Zhang and T. Zhao prove that, for \(N\geq2,\) \(P_M(t)\) has degree \(N-1\) and all its zeros are simple and negative [ZZ26, Thm. 1.1]. Thus \(A_M(t)\) has these zeros together with a simple zero at \(0.\) Their proof uses a forest decomposition, Lagrange inversion, and a factorization of the resulting differential operator into first-order factors.
The factorization also orders the associated gamma polynomials. If a composition \(\beta\) refines \(\alpha,\) then every nonconstant gamma coefficient and every ordered gamma zero weakly increases from \(P_\alpha\) to \(P_\beta\); all these inequalities are strict unless the two polynomials agree [ZZ26, Thm. 1.2]. This places every \(P_M\) between the Narayana and Eulerian polynomials in gamma-coefficient order, with analogous bounds on the gamma zeros.
Finally, adjoining \(r\) copies of a new maximal label gives a totally nonnegative coefficient transition matrix. Each of its column polynomials has \(r\) simple negative zeros, and adjacent columns strictly interlace [ZZ26, Thms. 1.3–1.4].
Example (Canon permutations).
A canon permutation of \(\{1^k,2^k,\dotsc,n^k\}\) is a multiset permutation for which the subsequence formed by the \(j\)th copy of every letter is the same permutation \(\sigma\in\symS_n\) for every \(j\in[k].\) Write \[C_n^k(t)\coloneqq\sum_{\pi}t^{\des(\pi)}.\] Elizalde proved the factorization \[C_n^k(t)=A_n(t)N_{n,k}(t), \qquad C_n^{k,\sigma}(t)=t^{\des(\sigma)}N_{n,k}(t),\] where \(N_{n,k}(t)=t^{1-n}W_{(k^n)}(t)\) is a rectangular Narayana polynomial [Eli25, Eq. (5) and Thm. 3.1]. Rectangular Narayana polynomials are real-rooted [CYZ18, Thm. 3.1], as are the Eulerian polynomials. Thus \(C_n^k(t)\) and every fixed-\(\sigma\) refinement \(C_n^{k,\sigma}(t)\) are real-rooted for all \(n,k\geq1.\)
This is separate from the colored-multiset theorem of Deligeorgaki, Han, and Solus above. Beck and Deligeorgaki’s labeled-poset explanation and the gamma-positivity result are discussed on the gamma-positivity page. The more general poset factorization there does not by itself imply real-rootedness. Likewise, the finer polynomials obtained by fixing a Dyck path need not be real-rooted; see [DM26, Remark 3.3].
#Specialized descent polynomials
Example (Parking-function descents).
The descent polynomial of parking functions is \[D_n(t) \coloneqq \sum_{a\in\mathrm{PF}_n}t^{\des(a)}.\] Its coefficients form A333829. For every \(n\geq 1,\) the polynomial \(D_n(t)\) is real-rooted.
Indeed, set \[W_{n,m}(t) \coloneqq \sum_{w\in[m]^n}t^{\des(w)}.\] The descent-set equidistribution of P. Diaconis and A. Hicks [DH17, Thm. 8] gives \[D_n(t)=\frac{1}{n+1}W_{n,n+1}(t).\] For the word side, let \(F_{r,j}(t)\) enumerate the words in \([m]^r\) ending in \(m+1-j.\) Appending the final letter gives \[F_{1,j}(t)=1,\qquad F_{r+1,j}(t) =t\sum_{i\lt{}j}F_{r,i}(t)+\sum_{i\geq j}F_{r,i}(t).\] The coefficient matrix has \(t\) below the diagonal and \(1\) on and above it; the matrix interlacing criterion shows that \(F_{r,1},\dotsc,F_{r,m}\) have a common interlacing. The compatibility theorem then implies that their sum \(W_{r,m}(t)\) is real-rooted. The transfer identity proves the claim.
Restricting to Smirnov words, whose adjacent letters differ, instead gives the real-rooted Chow polynomials of noncrossing partition lattices.
Example (Log-concavity from triangular-array recurrences).
U. Shankar gives sufficient conditions for log-concavity of rows of triangular arrays satisfying a super-recurrence generalizing the standard Eulerian, Stirling, and Lah recurrences [Sha25]. The proof uses a weighted-lattice-path interpretation and a weight-increasing injection, and it also introduces a two-parameter Eulerian analogue with palindromicity.
For the \(r=1\) specialization, Alexandersson settles the real-rootedness question. If \(\ell\geq1,\) set \(E^{(\ell)}_{1,0}=1,\) with entries outside the row equal to zero, and define \[E^{(\ell)}_{n,k}=(k+1)^\ell E^{(\ell)}_{n-1,k} +(n-k)^\ell E^{(\ell)}_{n-1,k-1},\] Then \(E_n^{(\ell)}(t)=\sum_kE^{(\ell)}_{n,k}t^k\) has only nonpositive real zeros and interlaces \(E_{n+1}^{(\ell)}(t).\) Its gamma polynomial is also real-rooted with nonnegative coefficients [Ale26, Thm. 5.1].
U. Shankar strengthens this result by constructing real-stable multivariate cross-step refinements for powered affine triangular recurrences [Sha26, Thms. B–C]. For the powered Eulerian specialization, the zeros of \(E_n^{(\ell)}(t)\) are simple when \(n\geq2,\) and the zeros of consecutive rows strictly interlace. The same paper interprets the \(\ell=2\) gamma coefficients through reduced synchronized pairs of increasing binary trees.
Example (Cyclic and even-top descents).
For northeast paths from \((0,0)\) to \((n,n),\) let \(\operatorname{cdes}(w)\) count cyclic occurrences of a north step followed by an east step. Then \[C_n(t)=\sum_wt^{\operatorname{cdes}(w)} =2\sum_{k=1}^n\binom nk\binom{n-1}{k-1}t^k.\] Its coefficients are A335340; \(C_n(t)\) has a simple zero at \(0\) and \(n-1\) simple negative zeros.
If \(R_N(t)\) enumerates permutations by descents whose top entry is even, then its coefficient triangle is A134434. For \(N\geq2,\) all zeros of \(R_N(t)\) are simple and negative, and consecutive polynomials strictly interlace [Ale26, Thms. 3.1 and 4.1].
Example (Rook-Eulerian polynomials).
Let \(\lambda=(\lambda_1,\dotsc,\lambda_n)\) be a Ferrers board with \(\lambda_i\geq i,\) and let \(Q_\lambda(t)\) enumerate row-complete nonattacking rook placements by ascents. The refinements by the column occupied in the first row form an interlacing sequence, so \(Q_\lambda(t)\) is real-rooted. The multivariate ascent-set enumerator is same-phase stable [AJQ25, Thms. 12–13]. For complete placements, these objects form a lower Bruhat interval below a \(312\)-avoiding permutation.
Example (Multi-dimensional permutations).
S. Chen, H. Fang, S. Kitaev, and Candice X. T. Zhang extend permutation-pattern statistics to multi-dimensional permutations [CFKZ25]. For canonical three-dimensional permutations, let \(R_{n,k}\) count the permutations with exactly \(k\) repeated levels under the max-level statistic, and set \(R_{n,*}(y) \coloneqq \sum_k R_{n,k}y^k.\) They prove that \(\{R_{n,*}(y)\}_{n\geq 0}\) is a generalized Sturm sequence, so each \(R_{n,*}(y)\) is real-rooted. They also show that weakly increasing three-dimensional permutations are counted by the Springer numbers.
Example (Eulerian polynomials with descents from prescribed set).
C. A. Athanasiadis, T. Douvropoulos, and K. Kalampogia-Evangelinou show that for any \(T \subseteq [n-1],\) the polynomials \[A_n^T(x) \coloneqq \sum_{\substack{\pi \in \symS_n \\ \DES(\pi) \subseteq T}} x^{\des(\pi)}\] are real-rooted [ADK24, Thm. 1.2]. These are restricted Eulerian polynomials. The proof refines the above polynomials by the value of \(\pi(n),\) and then uses a recursion which preserves an interlacing family of polynomials.
Example (Inversion-weighted Eulerian polynomials with fixed last entry).
Define the refinement \[A_{n,j}(t,q) \coloneqq \sum_{\substack{\sigma \in \symS_n \\ \sigma(n)=j}} t^{\des(\sigma)}q^{\inv(\sigma)} .\] For any \(q \gt 0,\) \[A_{n,n}(t,q), A_{n,n-1}(t,q), \dotsc, A_{n,1}(t,q)\] is an interlacing sequence, see [SV15]. A short proof is given in [AK23].
#Eulerian polynomials of classical types
Example (Type \(B\) Eulerian polynomials).
Now consider permutations of type \(B\). For \(\pi\in B_n,\) set \(\pi(0)\coloneqq 0,\) and let \(\des(\pi)\) be the cardinality of \[\{ i \in \{0,1,2,\dotsc,n-1\} : \pi(i) \gt \pi(i+1) \}.\] The type \(B\) Eulerian polynomials \(P_n(t)=\sum_{\sigma \in B_n} t^{\des(\sigma)}\) then satisfy the recurrence \[P_n(t) = ((2n-1)t+1) P_{n-1}(t) + 2t(1-t) P'_{n-1}(t),\] from which the interlacing property follows, see [Cho22]. They in fact show that the corresponding operator preserves real-rootedness, using the method of J. Borcea and P. Brändén on stable polynomials.
The analogous polynomials for type \(D\) are real-rooted, but consecutive type \(D\) polynomials are only conjectured to form interlacing pairs; see [Cho22, Section 5].
There is, however, a refined within-rank relation among the three classical types. Write \(A_n(t),\) \(B_n(t)=P_n(t),\) and \(D_n(t)\) for the Eulerian polynomials of types \(A,\) \(B,\) and \(D,\) respectively; in particular, \(D_n(t)\) is the type \(D\) Eulerian polynomial ; its coefficient triangle is A066094. For \(n\geq 2,\) Stembridge’s identity is \[D_n(t)=B_n(t)-n2^{n-1}tA_{n-1}(t).\]
Theorem (Ma, [Ma26, Thm. 1]).
For every \(n\geq 3,\) the type \(D\) Eulerian polynomial \(D_n(t)\) has only simple negative zeros. Moreover, the zeros of \(A_{n-1}(t),\) \(B_n(t),\) and \(D_n(t)\) interlace separately on \((-\infty,-1)\) and \((-1,0).\) The only break from ordinary interlacing occurs across the middle at \(-1\); this is called the half interlacing property.
The polynomial \(D_2(t)=(1+t)^2\) is the exceptional repeated-root case. When \(n\) is odd, all three polynomials have the common zero \(-1.\) This within-rank half interlacing is distinct from the open consecutive interlacing question for \(D_n(t)\) and \(D_{n+1}(t).\)
There is also a \(q\)-refinement. For an even signed permutation \(\sigma=(\sigma_1,\dotsc,\sigma_n),\) let \[\DES_D(\sigma)= \{0:\sigma_1+\sigma_2\lt{}0\} \cup\{i\in[n-1]:\sigma_i\gt{}\sigma_{i+1}\}, \qquad \operatorname{neg}_D(\sigma)=|\{i\geq2:\sigma_i\lt{}0\}|,\] and set \[D_n(t;q)=\sum_{\sigma\in D_n} q^{\operatorname{neg}_D(\sigma)}t^{|\DES_D(\sigma)|}.\] For every \(n\geq2\) and \(q\gt{}0,\) this polynomial is real-rooted. The proof uses a last-entry refinement whose transition matrix preserves mutual interlacing [YZ14, Thm. 3.8]. The same method proves that the affine Eulerian polynomial of every irreducible finite Weyl group is real-rooted [YZ14, Thm. 4.6].
#Binomial and inversion-sequence Eulerian polynomials
Example (Binomial Eulerian polynomials).
The binomial Eulerian polynomials can be defined as \[\tilde{A}_n(t) \coloneqq \sum_{\sigma \in \symS_n} (1+t)^{\fix(\sigma)} t^{\exc(\sigma)}.\] Athanasiadis conjectured that these are real-rooted, and it was later shown by J. Haglund and P.B. Zhang [HZ19] that this holds, by means of interlacing polynomials. Their result is stronger: it proves real-rootedness for a binomial version of the \(\svec\)-Eulerian polynomials defined below, and the case \(\svec=(2,3,\dotsc,n)\) recovers \(\tilde{A}_n(t)\) via Lehmer codes.
Computations suggest that \(\tilde{A}_n(t) \interl \tilde{A}_{n+1}(t),\) but this does not follow immediately from the reference above.
Example (Ascents and descents in inversion sequences).
Fix \(\svec = (s_1,\dotsc,s_n)\) and define \[\mathcal{I}_n^\svec \coloneqq \{ (e_1,\dotsc,e_n) \in \setZ^n : 0 \leq e_i \lt s_i,\; 1 \leq i \leq n \}\] with the conventions that \(e_0 = e_{n+1} =0\) and \(s_0 = s_{n+1}=1.\) This is the set of inversion sequences associated with \(\svec.\)
An index \(0 \leq i \leq n\) is an ascent, collision, or descent of an inversion sequence \(\evec\) if \[\frac{e_i}{s_i} \lt \frac{e_{i+1}}{s_{i+1}}, \qquad \frac{e_i}{s_i} = \frac{e_{i+1}}{s_{i+1}}, \qquad \frac{e_i}{s_i} \gt \frac{e_{i+1}}{s_{i+1}},\] respectively. Moreover, let \(\mathcal{D}_n^\svec \subset \mathcal{I}_n^\svec\) be the set of inversion sequences without collisions.
It was proved in [SV15] and [GS20], respectively, that \[E^\svec_n(t) = \sum_{\evec \in \mathcal{I}_n^\svec} t^{\asc(\evec)} \quad \text{and} \quad D^\svec_n(t) = \sum_{\evec \in \mathcal{D}_n^\svec} t^{\asc(\evec)}\] are real-rooted. The methods use the theory of interlacing sequences of polynomials. In particular, if \(\svec\) is a fixed sequence of positive integers, then \(E^\svec_n(t) \interl E^\svec_{n+1}(t).\)
The proof by Savage and Visontai also shows that the \(\svec\)-Eulerian polynomials are real-rooted. See also the proof in [Br15, Sec. 8.2].
J. Haglund and P.B. Zhang [HZ19] prove a binomial analogue of this theorem. Define the number of collisions by \[\operatorname{coll}(\evec) \coloneqq \left|\left\{0\leq i\leq n : \frac{e_i}{s_i}=\frac{e_{i+1}}{s_{i+1}}\right\}\right|.\] Their binomial \(\svec\)-Eulerian polynomial is \[\widetilde{E}_n^\svec(t) \coloneqq \sum_{\evec\in\mathcal{I}_n^\svec} (1+t)^{\operatorname{coll}(\evec)} t^{\asc(\evec)}.\] They prove that \(\widetilde{E}_n^\svec(t)\) is real-rooted for every positive integer sequence \(\svec.\) More precisely, after refining by the last entry \(e_m=k,\) the polynomials \[p_{m,0}^\svec(t),p_{m,1}^\svec(t),\dotsc,p_{m,s_m-1}^\svec(t)\] form an interlacing sequence for every \(m.\) The proof writes the recurrence from \(m-1\) to \(m\) as multiplication by a matrix whose entries are \(1,\) \(t,\) and \(1+t.\) They then use Brändén’s matrix criterion for interlacing preservers: it is enough to check the relevant \(2\times2\) submatrices. The \(1,t\)-only cases are known, and the new \(1+t\) cases are checked by factoring the small matrices into already known interlacing-preserving matrices, together with the convexity closure for interlacing sums.
#Stirling numbers and pattern statistics
Example (Type 1 Stirling numbers (cycles, left to right minima)).
The unsigned Stirling numbers of the first kind appear in the identity \[t(t+1)(t+2)\dotsm (t+n-1) = \sum_{k=1}^n |s(n,k)|t^k = \sum_{\pi \in \symS_n} t^{\mathrm{numberOfCycles}(\pi)} = \sum_{\pi \in \symS_n} t^{\mathrm{leftToRightMinima}(\pi)}\] where \(|s(n,k)|\) denotes the unsigned Stirling number of the first kind; see A132393.
Example (Higher-order Stirling triangles).
B. Deb and A. D. Sokal introduce higher-order Stirling cycle and subset triangles and formulate total-positivity and real-rootedness questions for them [DS25]. Their framework asks for total positivity of the triangle, its row reversal, Toeplitz total positivity of row sequences, and coefficientwise Hankel total positivity of the row-generating polynomials. Toeplitz total positivity is the Pólya frequency condition on each row sequence. They prove several cases for order \(2.\)
Example (Big descents in pattern-avoiding permutations).
A descent \(i\) of a permutation \(\pi\) is called a big descent if \(\pi_i\gt{}\pi_{i+1}+1.\) S. Elizalde, J. Rivera, and Y. Zhuang classify the big-descent distributions on permutations avoiding one or two patterns of length three [ERZ26]. Among their future directions, they prove real-rootedness for the big-descent polynomial on \(231\)-avoiding permutations and formulate real-rootedness, log-concavity, and Schur-positivity questions for the associated quasisymmetric functions.
#Peaks, runs, and derangements
#Peak and run polynomials
Example (Increasing runs in ternary words).
Let \[W_n(t)=\sum_{w\in\{0,1,2\}^n}t^{\operatorname{run}(w)},\] where \(\operatorname{run}(w)\) counts maximal strictly increasing consecutive factors. The coefficient triangle is A120987. Every \(W_n(t)\) is real-rooted, and \(W_n(t)\) and \(W_{n+1}(t)\) weakly interlace [Ale26, Thm. 6.1].
Example (Peak polynomials, peaks in permutations).
The peak-generating polynomials \[P_n(x)=\sum_{\sigma\in\symS_n} x^{\text{peaks}(\sigma)}\] have coefficients A008303. Real-rootedness was proved in [WS96], by using the recursion \[P_n(x) = (2+x(n-2)) P_{n-1}(x) + 2x(1-x) P'_{n-1}(x).\]
After setting \(Q_n=xP_n,\) the recursion becomes \[Q_n(x) = n x\cdot Q_{n-1}(x) + 2(1-x)x Q'_{n-1}(x).\]
The operator \(T_n:f \mapsto nx f + 2(1-x)x f'\) does not by itself give a real-rootedness preserver through the stable-operator criterion. However, the Ma–Wang theorem [MW08] applies to the recurrence, and gives that all roots of \(Q_n\) are real and that \(Q_n\interl Q_{n+1}.\)
More precisely, for \(n\geq3,\) \(P_n\) has \(\lfloor(n-1)/2\rfloor\) simple negative zeros, and consecutive polynomials strictly alternate. If \(\operatorname{pkv}(\sigma)\) is the set of values appearing at peaks, then \[\widehat P_n(\mathbf x)= \sum_{\sigma\in\symS_n}\prod_{j\in\operatorname{pkv}(\sigma)}x_j\] is stable; its consecutive positive weighted diagonal specializations interlace [Ale26, Thms. 6.2–6.3].
Example (Alternating runs).
S.-M. Ma and Y. Wang consider the alternating-run polynomial with coefficients A059427: \[P_n(x) = x \cdot \sum_{\sigma \in \symS_n} x^{\text{peaks}(\sigma)+\text{valleys}(\sigma)}.\] The polynomials satisfy the recursion \[P_n = x(nx-2x+2)P_{n-1} + x(1-x^2) P'_{n-1}.\] Ma and Wang’s general theorem gives the interlacing relation \(P_n \interl P_{n+1}\) here as well [MW08]. However, note that \(x=-1\) is a root with multiplicity \(\lfloor n/2 \rfloor -1.\) See [MW08] for more references—real-rootedness was observed earlier.
Example (Up-down runs and Stirling permutation codes).
Let \(T(n,k)\) count permutations in \(\symS_n\) with \(k\) up-down runs after a leading zero is adjoined; these numbers form A186370. If \[U_n(x)=\sum_kT(n,k)x^k, \qquad f_n(x)=\frac{2U_n(x)}x,\] then the Stirling-permutation-code expansion separates the odd and even run counts as \[f_n(x)=2\xi_n(x^2)+x\zeta_n(x^2),\] where \(\xi_n(t)=\sum_kT(n,2k+1)t^k\) and \(\zeta_n(t)=2\sum_kT(n,2k+2)t^k.\) The polynomials satisfy \[f_{n+1}(x)=(1+x+(n-1)x^2)f_n(x)+x(1-x^2)f'_n(x), \qquad f_1(x)=2.\] Every zero of \(f_n\) lies in \([-1,0),\) and \(f_n\) weakly interlaces \(f_{n+1}.\) More precisely, \(-1\) has multiplicity \(\lfloor n/2\rfloor\) and the remaining \(\lfloor(n-1)/2\rfloor\) zeros are simple and lie in \((-1,0).\) The polynomials \(\xi_n\) and \(\zeta_n\) also have only simple negative zeros; \(\zeta_{2m}\) alternates left of \(\xi_{2m},\) while \(\zeta_{2m+1}\) interlaces \(\xi_{2m+1}\) [MQYY24, Thm. 6].
#Descents on restricted permutations
Example (Descents in separable permutations).
Let \(\mathcal S_n\) be the separable permutations in \(\symS_n,\) and define \[S_n(t) \coloneqq \sum_{\pi\in\mathcal S_n}t^{\des(\pi)}.\] The coefficients of \(S_n(t)\) form row \(n\) of A175124. In his preprint [Zha26, Thm. 1.1], Z. Zhang proves that, for every \(n\geq2,\) the polynomial \(S_n\) has \(n-1\) simple negative zeros and strictly interlaces \(S_{n+1}.\) This resolves the separable-permutation part of the real-rootedness conjecture of S. Fu, Z. Lin, and J. Zeng [FLZ18].
We define the associated gamma-polynomial by \[S_n(t)=(1+t)^{n-1}\Gamma_n\!\left(\frac{t}{(1+t)^2}\right).\] For \(n\geq2,\) the polynomial \(\Gamma_n\) has degree \(\lfloor(n-1)/2\rfloor\) and simple negative zeros; consecutive gamma-polynomials strictly interleave [Zha26, Thm. 1.2].
Example (Descents and left peaks in simsun permutations).
Define the polynomials \[P_n(t)=\sum_{\sigma\in \mathrm{RS}_n} t^{\des(\sigma)},\] where the sum is over all simsun permutations. C.-O. Chow and W. C. Shiu prove that \[P_n(t) = ((n-1)t + 1) P_{n-1}(t) + t(1-2t) P'_{n-1}(t), \qquad P_0(t)=1.\] From this recursion, it follows that the \(P_n(t)\) are real-rooted and interlacing; see [CS11, Thm. 2.1 (iii)]. They also note that this is the same as counting left peaks in simsun permutations.
S.-M. Ma and Y.-N. Yeh later prove that interior peaks also give rise to real-rooted polynomials [MY16].
Example (Descents in run-sorted permutations).
In joint work with O. Nabawanda, we show that the run-sorted descent polynomials \(R_n(t)\) given by \(R_1(t)=R_2(t)=t\) and the recursion \[R_n(t) = t\cdot R'_{n-1}(t) + t\cdot (n-2) R_{n-2}(t),\] satisfy \(R_{n-1} \interl R_{n}\) for all \(n \geq 1\) [AN21]. The triangle of coefficients can be found in A124324.
The polynomial \(R_n(t)\) is the generating function for the number of runs in run-sorted permutations: \[R_n(t) = \sum_{\pi \in \text{RSP}(n)} t^{\des(\pi)+1}.\] These are therefore closely related to the classical Eulerian polynomials.
#Derangement polynomials
Example (Excedances and derangements, with proof).
The derangement polynomial is \(P_n(t)\coloneqq \sum_{\pi \in D(n)} t^{\exc(\pi)},\) where \(D(n)\) is the set of derangements (fixed-point-free permutations) of size \(n,\) see A046739. Then \(P_{n-1} \interl P_n.\)
The proof is as follows. From [MR03], one obtains \(P_1=0,\) \(P_2=t,\) and for \(n\geq 3,\) \[P_n = t \left( (n - 1) P_{n - 2} + (n-1) P_{n - 1} + (1-t) P'_{n - 1} \right).\] Note that \(P_n(t)\) is a polynomial of degree \(n-1,\) as the permutation \(\pi=2345\dotsc n 1\) contributes with \(t^{n-1}\) in the definition.
Proceed by induction over \(n,\) and assume that \(P_{n-2}\interl P_{n-1}\) for some \(n\geq 3.\)
By the induction hypothesis, and using that \(P'_{n-1} \interl P_{n-1},\) we can deduce that \[P_{n-2} \interl P_{n-1}, \text{ and } (n-1)P_{n-1} + (1-t)P'_{n-1} \interl P_{n-1}.\] Using [Wag92], we have \[\begin{aligned} (n-1)P_{n - 2} + (n-1)P_{n - 1} + (1-t)P'_{n-1} &\interl P_{n-1} \\ t\left((n-1)P_{n - 2} + (n-1)P_{n - 1} + (1-t)P'_{n-1}\right) &\interl t P_{n-1} \\ P_{n} &\interl t P_{n-1}. \end{aligned}\] It follows that \(P_{n-1}\interl P_n.\)
See also [LW07] where this is proved using a different recursion.
The descent distribution is a different normalization. Writing \(\mathcal{D}_n\) for the derangements in \(\symS_n,\) the polynomial \[D_n^{\des}(t)=\sum_{\pi\in\mathcal{D}_n}t^{\des(\pi)}\] has only nonpositive real zeros for every \(n\geq2\) [Ale26, Thm. 2.1].
#Set partitions
Example (Touchard polynomials (blocks in set partitions), with proof).
The Touchard polynomials may be defined via \[T_n(x) \coloneqq \sum_{P \in SP(n)} x^{\mathrm{blocks}(P)} = \sum_{k=1}^n S(n,k) x^k,\] where \(SP(n)\) is the set of set partitions of \([n],\) and we count the number of blocks. Note that the coefficients are Stirling numbers of the second type.
Wilf gives the recurrence [Wil94, pp. 20, 139] \[T_n(x) = x \cdot T_{n-1}(x) + x\cdot T'_{n-1}(x).\] We know \(T_{n-1} \interl x T_{n-1}.\) Moreover, \(T_{n-1} \interl x T'_{n-1},\) so by [Wag92], we have \(T_{n-1} \interl x T_{n-1} + x T'_{n-1}.\) But this is exactly \(T_{n-1} \interl T_n.\)
Another way to prove real-rootedness is by interpreting \(T_n(x)\) as a matching polynomial.
Example (Set partitions without singleton blocks).
Define \(D_n(t)=\sum_{\pi \in SP_2(n)} t^{\mathrm{blocks}(\pi)},\) where \(SP_2(n)\) is the set of set partitions of \([n]\) whose blocks all have size at least \(2.\) Then \[D_n(t) = t\left( D'_{n-1}(t) + (n-1) D_{n-2}(t) \right)\] and \(D_{n-1}(t)\) and \(D_n(t)\) have interlacing roots, see [BM16, Thm. 1]. Note that the degree of \(D_n(t)\) is \(\lfloor n/2 \rfloor.\)
The proof is short by using Wagner’s lemma and induction. The base case is immediate; assume \(D_{n-2} \interl D_{n-1}\) for some \(n \geq 2.\) We then know that \[D'_{n-1} \interl D_{n-1} \text{ and } (n-1)D_{n-2} \interl D_{n-1}.\] Wagner’s lemma now gives that \[D'_{n-1} + (n-1)D_{n-2} \interl D_{n-1} \implies D_{n-1} \interl t \left( D'_{n-1} + (n-1)D_{n-2} \right),\] and this is precisely \(D_{n-1} \interl D_n,\) which completes the induction step.
This result also follows from the general machinery in [Hit24, Subsec. 3.4]. A recursion for set partitions into blocks of size at least \(s\) is also mentioned there (originally due to L. Comtet) but it does not immediately imply real-rootedness.
Example (Type \(B\) set partitions).
D. Wang considers a type \(B\) version, and more generally a colored version, of counting blocks in set partitions [Wan14]. The polynomials considered satisfy the (general) recursion \[T_n(x) = (x+c) \cdot T_{n-1}(x) + mx\cdot T'_{n-1}(x),\] where \(m\) and \(c\) are positive integers.
P. Alexandersson, F. Beyene, and R. Mantaci study type \(B\) set partitions without a zero block [ABM26]. If \(b_{n,k}\) counts the merging-free ones with \(k\) blocks and \(M_n(t)=\sum_k b_{n,k}t^k,\) then \[M_n(t)=2tM'_{n-1}(t)+2(n-2)tM_{n-2}(t),\qquad n\geq2,\] with \(M_0(t)=1\) and \(M_1(t)=t.\) The polynomials \(M_n(t)\) are real-rooted and consecutive members interlace [ABM26, Prop. 27 and Cor. 28]; the paper also proves real-rootedness for the block polynomials of type \(B\) partitions that are both merging-free and separated [ABM26, Thm. 33]. Its flattened signed-permutation descent polynomial is gamma-positive and real-rooted [ABM26, Thm. 59 and Cor. 60].
Example (Ordered partitions).
Simion considers any multiset \(M\) and the number \(O(M,k)\) of compositions into exactly \(k\) parts [Sim84]. The polynomial \(\sum_k O(M,k)x^k\) is real-rooted.
In particular, this implies the real-rootedness of the Touchard polynomials.
#Stirling and quasi-Stirling permutations
Example (Stirling permutations).
The descent number of a Stirling permutation \(\pi \in Q_n\) is the cardinality of \[\{ i : \pi_i \gt \pi_{i+1} \} \cup \{2n\}.\] The Stirling permutation descent polynomials satisfy the recurrence \[P_n(t) = (2n+1)t \cdot P_{n-1}(t) + t(1-t) P'_{n-1}(t),\] see [Car65, Eq. 13] and [GS78, Thm. 2.1]. The coefficients can be found in A008517; the first few values are presented below.
\( 1 \) \( 1 \) \( 2 \) \( 1 \) \( 8 \) \( 6 \) \( 1 \) \( 22 \) \( 58 \) \( 24 \) \( 1 \) \( 52 \) \( 328 \) \( 444 \) \( 120 \) \( 1 \) \( 114 \) \( 1452 \) \( 4400 \) \( 3708 \) \( 720 \)These numbers appear in the Hilbert series of the Whitehouse complex; see [Rea05]. These polynomials are the \(h^*\)-polynomials of the order polytope defined by the inequalities \[y_1 \leq y_2 \leq \dotsb \leq y_n \qquad 0 \leq y_j \leq x_j \leq 1 \text{ for } j=1,\dotsc,n.\] S.-M. Ma, J. Ma, J. Yeh, and Y.-N. Yeh prove \(e\)-positivity for multivariate \(k\)th-order Eulerian polynomials attached to \(k\)-Stirling permutations [MMYY21]. They also give tree interpretations for the coefficients in related gamma-type expansions. For \(n=20,\) the linear term of the Ehrhart polynomial (of degree \(40\)) has negative coefficient \(-22354369153/6983776800.\)
M. Bóna proved that these have only real zeros [Bo09]. The Ma–Wang theorem also gives real-rootedness and interlacing.
J. Haglund and M. Visontai [HV12, Thm. 3.3] show that a multivariate generalization is stable.
Theorem (Dey’s interlacing criterion, [Dey23]).
Let \((F_n(x))_{n\geq 0}\) be a sequence of polynomials satisfying \[F_n(x)=\bigl(\alpha_1(n)x+\alpha_2\bigr)F_{n-1}(x) +\alpha_3x(1-x)F'_{n-1}(x),\] where \(\alpha_2,\alpha_3\in\setN\cup\{0\}.\) Suppose that each \(F_n(x)\) is real-rooted with distinct nonpositive zeros, and that some integer \(k\) satisfies, for all \(n\geq k,\) \[2\alpha_1(n)-n\alpha_3\gt{}0, \qquad n\alpha_1(n)\alpha_3+3n\alpha_2\alpha_3 -4\alpha_1(n)\alpha_2\gt{}0.\] Then \(F_{n-1}(x)\interl F_n(x)\) for all \(n\geq k.\)
Dey applies this criterion to the type \(A\) and type \(B\) Eulerian polynomials, to the Stirling permutation polynomials above, and to restricted Eulerian polynomials. Thus it is a useful way to upgrade an already known real-rootedness proof to a consecutive interlacing statement when the recurrence has the form above.
#Jacobi–Stirling descent polynomials
For \(k\geq1,\) we order the multiset \[M_k=\{\overline{1},1,1,\overline{2},2,2,\dotsc,\overline{k},k,k\} \quad\text{by}\quad \overline{1}\lt{}1\lt{}\overline{2}\lt{}2\lt{}\dotsb\lt{}\overline{k}\lt{}k.\] For \(S\subseteq[k],\) delete the barred letters \(\overline{j}\) with \(j\in S.\) A Jacobi–Stirling permutation of the resulting multiset is a word in which every entry between the two unbarred copies of \(j\) is larger than \(j,\) for each \(j\in[k].\) Let \(\operatorname{JSP}_{k,S}\) be the set of these words. We define \[A_{k,S}(t)\coloneqq\sum_{w\in\operatorname{JSP}_{k,S}}t^{\des(w)}, \qquad A_{k,i}(t)\coloneqq\sum_{\substack{S\subseteq[k]\\|S|=i}}A_{k,S}(t), \qquad 0\leq i\leq k,\] where \(\des(w)=|\{j:1\leq j\lt{}|w|,\ w_j\gt{}w_{j+1}\}|\) counts internal descents. In particular, \(A_{k,k}(t)\) is the second-order Eulerian polynomial for Stirling permutations, with the final descent omitted. Its coefficients are row \(k\) of A008517: \([t^j]A_{k,k}(t)=T(k,j+1)\) for \(0\leq j\leq k-1.\) Gessel, Lin, and Zeng observed that each fixed-\(S\) polynomial is real-rooted and conjectured real-rootedness of \(A_{k,i}(t)\) for all \(k\) and \(i\) [GLZ12, Section 5 and Conjecture 15]. Their normalization includes an additional factor of \(t.\)
Theorem (S.-M. Ma and M.-X. Wang, [MW26, Thm. 2 and Cor. 2]).
Let \(k\geq1\) and \(1\leq i\leq k,\) with \[i\in\{1,2,3,k-2,k-1\}.\] For any nonnegative real weights \(c_S,\) not all zero, the polynomial \[F_{k,i}(t)\coloneqq \sum_{\substack{S\subseteq[k]\\|S|=i}}c_S A_{k,S}(t)\] has only simple negative zeros. In particular, this holds for \(A_{k,i}(t).\)
The five families correspond to deleting one, two, or three barred letters, or retaining one or two. The boundary cases \(i=0\) and \(i=k\) were already known; the conjecture for arbitrary \(i\) remains open.
The proof uses the insertion operator \[D_Lf(t)\coloneqq(1+Lt)f(t)+t(1-t)f'(t),\] where \(L\) is the word length before insertion. With \(H_k=D_{3k-3}A_{k-1,0}\) for \(k\geq2,\) the weighted polynomials above satisfy \[\begin{aligned} F_{k,1}&\interl A_{k,0}, & F_{k,2}&\interl H_k, & F_{k,3}&\interl A_{k-1,0},\\ A_{k,k}&\interl F_{k,k-1}, & A_{k+1,k+1}&\interl F_{k,k-2}. \end{aligned}\] Each relation holds in the corresponding parameter range of the theorem, and all interlacing inequalities are strict [MW26, Cor. 2].
Theorem ([MW26, Thms. 3 and 5]).
For \(k\geq2\) and \(0\leq i\leq k,\) write \(A_{k,i}(t)=\sum_{j=0}^{d}a_jt^j,\) where \(d=2k-i-1.\) Then \[a_j\lt{}a_{d-j}\quad(0\leq j\lt{}d/2), \qquad a_0\lt{}a_1\lt{}\dotsb\lt{}a_{\lfloor d/2\rfloor}.\] Thus the coefficient sequence is strictly top heavy and has an increasing left half. The same inequalities hold for every \(A_{k,S}(t),\) with \(d=2k-|S|-1.\)
#Quasi-Stirling descent polynomials
Example (Quasi-Stirling permutations).
A quasi-Stirling permutation of size \(n\) is a permutation of the multiset \(\{1,1,2,2,\dotsc,n,n\}\) avoiding the patterns \(1212\) and \(2121.\) Let \(\widetilde{Q}_n\) be the set of quasi-Stirling permutations of size \(n,\) and set \[\widetilde{Q}_n(t) \coloneqq \sum_{\pi \in \widetilde{Q}_n} t^{\des(\pi)},\] where the final position is counted as a descent, as above.
S. Elizalde [Eli21, Thm. 3.2] proved that \(\widetilde{Q}_n(t)\) has real, distinct, and nonpositive roots. More generally, for \(1 \leq r \leq n,\) let \[A_{n,r}(t) \coloneqq \sum_{\pi \in \symS_n} t^{\exc_r(\pi)}, \qquad J_{n,r}(t) \coloneqq \sum_{\pi \in \operatorname{Inj}([n-r],[n])} t^{\exc(\pi)},\] where \(\exc_r(\pi) \coloneqq |\{i:\pi_i \geq i+r\}|.\) Then \(A_{n,r}(t)\) and \(J_{n,r}(t)\) also have real, distinct, and nonpositive roots. The quasi-Stirling case follows from the relation \[\widetilde{Q}_n(t)=t J_{2n,n+1}(t).\] Elizalde also proves asymptotic normality for the descent distribution and extends the analysis to a one-parameter family generalizing \(k\)-Stirling permutations.
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