# conjectures

This is a compilation of conjectures and open questions mainly from research articles with updates on their current status. If the information or attributions are not correct please contact me.

**Legend:** open, ~~settled~~

- (2/24/2018) The famous Higman conjecture states that the number k(U_n(F_q)) of conjugacy classes of the group U_n(F_q) of n by n upper triangular matrices over a finite field F_q is a polynomial in q. This conjecture has been verified up to n=16 by Pak and Soffer [1]. Also, Halasi and Pálfy [2] showed that Higman's conjecture fails for other class of groups defined from partially ordered sets. The data for k(U_n(F_q)) gives nonnegative polynomials in T=q-1. Kirilov conjectured that for q=2,3 the sequence {k(U_n(F_q))} is at least the Euler numbers A_n and the Springer numbers S_n. Computations with the data in [1] suggest the following more general conjecture (Kirilov's conjecture is included when T=1, 2):
**Conjecture:**Let T=q-1 and A_n(T) be defined as the coefficients of the following exponential generating function 1/(1-sin(Tx))^(1/T). Then for an infinite family of primes such that k(U_n(F_q)) is a polynomial in q then k(U_n(F_(T+1)) - A_n(T) is a polynomial with nonnegative coefficients.

## From papers

**A combinatorial model for computing volumes of flow polytopes, **with Carolina Benedetti, Rafael S. González D'León, Christopher R. H. Hanusa, Pamela E. Harris, Apoorva Khare, and Martha Yip, 2018, arxiv:1801.07684

- Conjecture 6.16: The volume of the Caracol flow polytope of size n with netflow (a,b,c,c...,c) is Cat(n-2)*a^(n-2)*(a+(n-1)*b)*(a+b+(n-2)*c)^(n-3)
~~Conjecture 8.7:~~Product formula for the polynomial E_PS_(n+1)(x) for the Pitman-Stanley graph (see paper for explicit formula). This follows from known facts of lattice points of the Pitman-Stanley polytope (proof will be included in arxiv v.2 o the paper above)- Conjecture 8.8: Product formula for the polynomial E_Car_(n+1)(x) for the Caracol graph (see paper for explicit formula).

**Rook theory of the finite general linear group, **with Joel Brewster Lewis, 2017, arxiv:1707.08192,

- Conjecture 6.3: Given any board B ⊆ [m] × [n], rank r, and prime power q, the q-hit number Hr(B, q) is nonnegative.
- Question 6.5: For which permutations w do the q-hit numbers Hi(bar(Iw), q) have positive coefficients?
- Conjecture 6.8: For the permutation v = (2n−1)(2n)(2n−3)(2n−2)· · · 3412 we have that the number of 2n by 2n invertible matrices over Fq with support in the complement of the inversions of v is in N[q].
- Conjecture 6.9: For every 123-avoiding permutation w of size n we have that the number of n by n invertible matrices over Fq with support in the complement of the inversions of w is in N[q].

**Hook formulas for skew shapes III. Multivariate and product formulas, **with Igor Pak and Greta Panova, 2017, arxiv:1707.00931

~~Conjecture 5.17~~**Theorem**(joint with C. Krattenthaler): Let λ = (2a+c) c+a (a+c) a , µ = (a+ 1)a a−11, then the number of standard Young tableaux of shape λ / µ equals a product formula (see paper for formula).- proved by Jang Soo Kim and Meesue Yoo in arxiv:1806.01525

~~Conjecture 9.6:~~**Theorem**a product formula for the number standard tableaux of a 4-parameter shifted skew shape Λ(a, c, d, m) (see paper for formula).- proved by Jang Soo Kim and Meesue Yoo in arxiv:1806.01525

**Flow polytopes and the space of diagonal harmonics, **with Ricky I. Liu and Karola Mészáros, 2016, arxiv:1610.08370, accepted *Canadian Journal of Mathematics*

- Conjecture 6.1: Let G be a threshold graph with n + 1 vertices. Then the q,t Ehrhart series of the flow polytope F_G with netflow (n,-1,-1,...,-1) is a q,t positive polynomial. (verified up to n=9)
- Conjecture 6.3: Let G be a threshold graph with n + 1 vertices. Then the difference of q,t Ehrhart series of the flow polytope F_(K_(n+1))(n,-1,...,-1) and F_G(n,-1,...,-1) is q,t positive. (verified up to n=9)
- Conjecture 6.4: For threshold graphs G and H with n+1 vertices, H sugraph of G we have that the difference of q,t Ehrhart series of the flow polytope F_G(n,-1,...,-1) and F_H(n,-1,...,-1) is q,t positive. (verified up to n=9)

**Asymptotics for the number of standard Young tableaux of skew shape, **with Igor Pak and Greta Panova, 2016, arxiv:1610.07561, to appear *European Journal of Combinatorics, *

~~Section 13.7~~: For the thick ribbons (2k-1,2k-2,...,1)/(k-1,k-2,...1) the number of SYT of this skew shape equals (n/2)log n + cn + o(n) for some c<0 and n = binomial(2k,2)-binomial(k,2).- settled in joint work with Igor Pak and Martin Tassy: arxiv:1805.00992

**Hook formulas for skew shapes II. Combinatorial proofs and enumerative applications, **with Igor Pak and Greta Panova, 2016, arxiv:1610.04744, *SIAM Journal of Discrete Math (SIDMA),* Vol 31 (2017), 1953–1989.

- Conjecture 7.9. If (θ1, . . . , θk) is a Lascoux–Pragacz decomposition of λ/µ ⊂ d × (n − d), then s_µ (x | y) · s_λ (x | y)^(k-1) = det [ s_λ\ θi#θj (x | y) ] where λ \ θi#θj denotes the partition obtained by removing from λ the outer substrip θi#θj . s_λ (x | y) denotes a factorial Schur function.
~~Conjecture 9.3:~~**Theorem:**The number of pleasant diagrams of the shape (n+2k-1,n+2k-2,...,1)/(n-1,n-2,...,1) is given by a certain determinant of Schroder numbers.- Proved by Byung-Hak Hwang, Jang Soo Kim, Meesue Yoo, Sun-mi Yun in arxiv:1711.02337

~~Conjecture 9.6:~~**Theorem:**determinantal formula for generating function of reverse plane partitions of skew shape (n+2k-1,n+2k-2,...,1)/(n-1,n-2,...,1). Proved independently by- Byung-Hak Hwang, Jang Soo Kim, Meesue Yoo, Sun-mi Yun in arxiv:1711.02337
- Peter L. Guo, C.D. Zhao, Michael X.X. Zhong in arxiv:1711.03048

**Probabilistic trees with algebraic roots, **with Olivier Bernardi, *Electronic Journal of Combinatorics*, Vol 23, No 2 (2016), arXiv:1501.01135, code

- Conjecture 21: (see paper for details)

**On flow polytopes, order polytopes, and certain faces of the alternating sign matrix polytope, **with Karola Mészáros and Jessica Striker, 2015, arXiv:1510.03357

- Remark 3.9: If G is a planar directed graph then a result of Postnikov shows that the flow polytope F_G is equivalent to an order polytope. Is the Chan-Robbins-Yuen polytope, the flow polytope of a complete graph Kn, an order polytope? The answer is positive up to n=6.

**Combinatorics of diagrams of permutations, **with Joel Brewster Lewis, arXiv:1405.1608, *Journal of Combinatorial Theory Series A, *Vol 137 (2016), 273-306; supplementary code and data

- Conjecture 4.11: For a permutation w of size n avoiding the patterns 4231, 35142, 42513, and 351624 we have that generating polynomial of pseudo Le and pseudo Gamma diagrams of w equal the Poincare polynomial q^(ell(w)) P_w(1/q). (verified up to n=7)

**Structure and enumeration of (3+1)-free posets, **with Mathieu Guay-Paquet and Eric Rowland, *Annals of Combinatorics*, Vol 18, No 4 (2014), 645-674, arXiv:1303.3652, related extended abstract FPSAC France 2013

- Question 1: The generating functions of the number of 2+2 free posets and of the number of 3+1 free posets are not D-finite. Are they ADE?
- Question 2: The generating function of the number of 2+2 and 3+1 free posets (counted by the Catalan numbers) is D-finite. What is the classification of the generating functions of the number of a+b free posets and (a+b, c+d) free posets?

**Flow polytopes of signed graphs and the Kostant partition function, **with Karola Mészáros, 2012, arXiv:1208.0140*, International Mathematics Research Notices,* No 3 (2015), 830-871.

~~Conjecture 7.12 (a).~~**Theorem:**The normalized volume of the type D Chan-Robbins-Yuen polytope of order n equals 2^((n-1)^2)*Cat(1)*...*Cat(n-1). Proved independently by- Doron Zeilberger in arxiv:1407.2829
- Jang Soo Kim in arxiv:1407.3467

~~Conjecture 7.12 (b).~~**Theorem:**The normalized volume of the type C Chan-Robbins-Yuen polytope of order n equals equals 2^(n(n-1))*Cat(1)*...*Cat(n-1).- proved by Sylvie Corteel, Jang Soo Kim, Karola Mészáros in arxiv:1704.02701

**Counting matrices over finite fields with support on skew Young diagrams and complements of Rothe diagrams, **with Aaron J. Klein and Joel Brewster Lewis, *Journal of Algebraic Combinatorics*, Vol 39, No 2 (2014) 429-456, arxiv:1203.5804

~~Conjecture 5.1.~~If Rw is the Rothe diagram of a permutation w in Sn and 0 ≤ r ≤ n then the number matq(n, Rw, r)/(q − 1) of n by n rank r matrices with support in the complement of Rw divided by (q-1)^r is a polynomial in q with nonnegative integer coefficients.- polynomiality and positivity when w avoids 1324, 24153, 31524, and 426153. Joint work with Joel Lewis, arxiv:1405.1608,
- polynomiality settled for all permutations, positivity is
*not true*in general (first counterexample at n=r=10). Joint work with Joel Lewis, arxiv:1707.08192.

**Matrices with restricted entries and q-analogues of permutations, **with Joel Brewster Lewis, Ricky I. Liu, Greta Panova, Steven V Sam, Yan Zhang, *Journal of Combinatorics*, Vol 2, No 3 (2012) 355-396, arXiv:1011.4539;

~~Question 5.6~~. Computations suggest that the number of invertible matrices over F_q, divided by (q-1)^n, with:~~(a)~~support on a skew shape yields a polynomial in q with nonnegative coefficients.- Settled in joint work with Aaron Klein and Joel Lewis: arXiv:1203.5804

- (b) support on complement of a skew shape, the number appears to be a polynomial.
- Settled in joint work with Joel Lewis, arxiv:1707.08192.

~~Question 5.7~~. Previous results suggest similarities between the number of invertible matrices over F_q with support in S and support in the complement of S that is reminiscent of the classical reciprocity of rook placements and rook numbers. Does this reciprocity hold in more generality?- Settled by Alberto Ravagnani in arxiv:1510.02383 using coding theory, further results developed in joint work with Joel Lewis arxiv:1707.08192.

## From phd thesis

**Combinatorics of colored factorizations, flow polytopes, and matrices over finite fields**, thesis, MIT 2012** **

- Conjecture 2.3.8. There appears to be symmetry of colored factorizations for two or three cycles. Note: examples show there is no such symmetry for four cycles.