On total positivity of Catalan-Stieltjes matrices. (English) Zbl 1351.05041
Summary: Recently X. Chen et al. [Linear Algebra Appl. 471, 383–393 (2015; Zbl 1307.05020)] proved some sufficient conditions for the total positivity of Catalan-Stieltjes matrices. Our aim is to provide a combinatorial interpretation of their sufficiant conditions. More precisely, for any Catalan-Stieltjes matrix \(A\) we construct a digraph with a weight, which is positive under their sufficient conditions, such that every minor of \(A\) is equal to the sum of weights of families of nonintersecting paths of the digraph. We have also an analogue result for the minors of Hankel matrix associated to the first column of Catalan-Stieltjes matrix \(A\).
MSC:
| 05B20 | Combinatorial aspects of matrices (incidence, Hadamard, etc.) |
| 15B36 | Matrices of integers |
| 15A45 | Miscellaneous inequalities involving matrices |
Citations:
Zbl 1307.05020References:
| [1] | M. Aigner, Catalan-like numbers and determinants, J. Combin. Theory Ser. A 87 (1999), no. 1, 33-51. · Zbl 0929.05004 |
| [2] | M. Aigner, A course in enumeration, Springer Berlin Heidelberg New York, 2007. · Zbl 1123.05001 |
| [3] | M. Aigner, A characterization of the Bell numbers, Discrete Math. 205 (1999) 207– 210. · Zbl 0959.11014 |
| [4] | J. Bonin, L. Shapiro, R. Simion, Some q-analogues of the Schr¨oder numbers arising from combinatorial statistics on lattice paths, J. Statist. Plann. Inference 34 (1993), no. 1, 35-55. · Zbl 0783.05008 |
| [5] | F. Brenti, Combinatorics and total positivity, J. Combin. Theory A 71 (1995): 175– 218. · Zbl 0851.05095 |
| [6] | F. Brenti, The applications of total positivity to combinatorics, and conversely, in: Total Positivity and Its Applications, Jaca, 1994, in: Math. Appl., vol. 359, Kluwer, Dordrecht, 1996, pp. 451-473. · Zbl 0895.05001 |
| [7] | X. Chen, H. Liang, Y. Wang, Total positivity of recursive matrices, Linear Algebra Appl. 471 (2015) 383-393. · Zbl 1307.05020 |
| [8] | Shaun M. Fallat, Charles R. Johnson, Hadamard powers and totally positive matrices, Linear Algebra Appl. 423 (2007), no. 2-3, 420-427. · Zbl 1123.15014 |
| [9] | Shaun M. Fallat, Charles R Johnson, Totally nonnegative matrices, Princeton Series in Applied Mathematics. Princeton University Press, Princeton, NJ, 2011. · Zbl 1390.15001 |
| [10] | P. Flajolet, Combinatorial aspects of continued fractions, Discrete Math. 32 (1980) 125-161. · Zbl 0445.05014 |
| [11] | S. Fomin, A. Zelevinsky, Total Positivity: Tests and parametrizations, Math. Intelligencer, Vol. 22, Issue 1(2000), pp. 23-33. · Zbl 1052.15500 |
| [12] | I. Gessel, G. Viennot, Binomial determinants, paths, and hook length formulae. Adv. in Math. 58 (1985), no. 3, 300-321. the electronic journal of combinatorics 23(4) (2016), #P4.3317 · Zbl 0579.05004 |
| [13] | S. Karlin, G. McGregor, Coincidence probabilities, Pacific J. Math. 9 (1959) 1141– 1164. · Zbl 0092.34503 |
| [14] | H. Kim, R. P. Stanley, A refined enumeration of hex trees and related polynomials, European. J. Combin. 54 (2016) 207-219. · Zbl 1331.05109 |
| [15] | H. Liang, L. Mu, Y. Wang, Catalan-like numbers and Stieltjes moment sequences, Discrete. Math. 339 (2016) 484-488. · Zbl 1327.05019 |
| [16] | B. Lindstr¨om, On the vector representations of induced matroids, Bull. London Math. Soc., 5 (1973) 85-90. · Zbl 0262.05018 |
| [17] | OEIS Foundation Inc. (2011), The On-Line Encyclopedia of Integer Sequences, http://oeis.org/A002212 |
| [18] | P. Mongelli, Total positivity properties of Jacobi-Stirling numbers, Adv. in Appl. Math. 48 (2012), no. 2, 354-364. · Zbl 1237.05007 |
| [19] | A. Pinkus, Totally positive matrices, Cambridge University Press, Cambridge, 2010. · Zbl 1185.15028 |
| [20] | L. W. Shapiro, A Catalan triangles, Discrete Math. 14 (1976) 83-90. · Zbl 0323.05004 |
| [21] | B. E. Sagan, Unimodality and the reflection principle, Ars Combin. 48 (1998) 65-72. · Zbl 0962.05003 |
| [22] | Alan Sokal, Total positivity: A concept at the interface between algebra, analysis and combinatorics, talk given at Institut Camille Jordan, Universit´e Lyon 1, April 20 2015. |
| [23] | T.-J. Stieltjes, Sur la r´eduction en fraction continue d’une s´erie proc´edant suivant les puissances descendantes d’une variable, Reprint of the 1889 original. Ann. Fac. Sci. Toulouse Math. (6) 5 (1996), no. 1, H1-H17. · Zbl 0869.40002 |
| [24] | H. S. Wall, Analytic theory of continued fractions, D. Van Nostrand Company, Inc., New York, N. Y., 1948. · Zbl 0035.03601 |
| [25] | Charles Zhao-Chen Wang, Yi Wang, Yi, Total positivity of Catalan triangle, Discrete Math. 338 (2015), no. 4, 566-568. · Zbl 1305.05004 |
| [26] | G. Viennot, Une th´eorie combinatoire des polynˆomes orthogonaux g´en´eraux, Lecture Notes at LACIM, Universit´e du Qu´ebec ‘a Montr´eal, 1983. |
| [27] | Yi Wang, Bao-Xuan Zhu, Log-convex and Stieltjes moment sequences, Adv. in Appl. Math., 81 (2016), 115-127. · Zbl 1352.05034 |
| [28] | Bao-Xuan Zhu, Log-convexity and strong q-log-convexity for some triangular arrays, Adv. in Appl. Math., 50 (2013), no. 4, 595-606. · Zbl 1277.05014 |
| [29] | Bao-Xuan Zhu, Some positivities in certain triangular arrays. Proc. Amer. Math. Soc. 142 (2014), no. 9, 2943-2952. · Zbl 1305.05020 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
