Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 002, 172 pages      arXiv:1502.00426      https://doi.org/10.3842/SIGMA.2016.002

On Some Quadratic Algebras I $\frac{1}{2}$: Combinatorics of Dunkl and Gaudin Elements, Schubert, Grothendieck, Fuss-Catalan, Universal Tutte and Reduced Polynomials

Anatol N. Kirillov abc
a) Research Institute of Mathematical Sciences (RIMS), Kyoto, Sakyo-ku 606-8502, Japan
b) The Kavli Institute for the Physics and Mathematics of the Universe (IPMU), 5-1-5 Kashiwanoha, Kashiwa, 277-8583, Japan
c) Department of Mathematics, National Research University Higher School of Economics, 7 Vavilova Str., 117312, Moscow, Russia

Received March 23, 2015, in final form December 27, 2015; Published online January 05, 2016

Abstract
We study some combinatorial and algebraic properties of certain quadratic algebras related with dynamical classical and classical Yang-Baxter equations.

Key words: braid and Yang-Baxter groups; classical and dynamical Yang-Baxter relations; classical Yang-Baxter, Kohno-Drinfeld and $3$-term relations algebras; Dunkl, Gaudin and Jucys-Murphy elements; small quantum cohomology and $K$-theory of flag varieties; Pieri rules; Schubert, Grothendieck, Schröder, Ehrhart, Chromatic, Tutte and Betti polynomials; reduced polynomials; Chan-Robbins-Yuen polytope; $k$-dissections of a convex $(n+k+1)$-gon, Lagrange inversion formula and Richardson permutations; multiparameter deformations of Fuss-Catalan and Schröder polynomials; Motzkin, Riordan, Fine, poly-Bernoulli and Stirling numbers; Euler numbers and Brauer algebras; VSASM and CSTCPP; Birman-Ko-Lee monoid; Kronecker elliptic sigma functions.

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References

  1. Aguiar M., On the associative analog of Lie bialgebras, J. Algebra 244 (2001), 492-532.
  2. Albenque M., Nadeau P., Growth function for a class of monoids, in 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009), Discrete Math. Theor. Comput. Sci. Proc., AK, Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 2009, 25-38.
  3. Arnold V.I., The cohomology ring of the colored braid group, Math. Notes 5 (1969), 138-140.
  4. Astashkevich A., Sadov V., Quantum cohomology of partial flag manifolds $F_{n_1\cdots n_k}$, Comm. Math. Phys. 170 (1995), 503-528, hep-th/9401103.
  5. Bardakov V.G., The virtual and universal braids, Fund. Math. 184 (2004), 1-18, math.GR/0407400.
  6. Bar-Natan D., Vassiliev and quantum invariants of braids, in The Interface of Knots and Physics (San Francisco, CA, 1995), Proc. Sympos. Appl. Math., Vol. 51, Amer. Math. Soc., Providence, RI, 1996, 129-144, q-alg/9607001.
  7. Bartholi L., Enriquez B., Etingof P., Rains E., Groups and algebras corresponding to the Yang-Baxter equations, J. Algebra 305 (2006), 742-764, math.RA/0509661.
  8. Bazlov Yu., Nichols-Woronowicz algebra model for Schubert calculus on Coxeter groups, J. Algebra 297 (2006), 372-399, math.QA/0409206.
  9. Belavin A.A., Drinfeld V.G., Triangle equations and simple Lie algebras, Classic Reviews in Mathematics and Mathematical Physics, Vol. 1, Harwood Academic Publishers, Amsterdam, 1998.
  10. Benson B., Chakrabarty D., Tetali P., $G$-parking functions, acyclic orientations and spanning trees, Discrete Math. 310 (2010), 1340-1353, arXiv:0801.1114.
  11. Berget A., Products of linear forms and Tutte polynomials, European J. Combin. 31 (2010), 1924-1935, arXiv:0906.4774.
  12. Bershtein M., Dotsenko V., Khoroshkin A., Quadratic algebras related to the bi-Hamiltonian operad, Int. Math. Res. Not. 2007 (2007), rnm122, 30 pages, math.RA/0607289.
  13. Billey S.C., Jockusch W., Stanley R.P., Some combinatorial properties of Schubert polynomials, J. Algebraic Combin. 2 (1993), 345-374.
  14. Birman J.S., Brendle T.E., Braids: a survey, in Handbook of Knot Theory, Elsevier B.V., Amsterdam, 2005, 19-103, math.GT/0409205.
  15. Birman J.S., Ko K.H., Lee S.J., A new approach to the word and conjugacy problem in the braid groups, Adv. Math. 139 (1998), 322-353, math.GT/9712211.
  16. Blasiak J., Liu R.I., Mészáros K., Subalgebras of the Fomin-Kirillov algebra, arXiv:1310.4112.
  17. Bourbaki N., Lie groups and Lie algebras, Chapters 4-6, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 2002.
  18. Bressoud D.M., Proofs and confirmations. The story of the alternating sign matrix conjecture, MAA Spectrum, Mathematical Association of America, Washington, DC, Cambridge University Press, Cambridge, 1999.
  19. Brualdi R., Kirkland S., Aztec diamonds and digraphs, and Hankel determinants of Schröder numbers, J. Combin. Theory Ser. B 94 (2005), 334-351.
  20. Chan C.S., Robbins D.P., On the volume of the polytope of doubly stochastic matrices, Experiment. Math. 8 (1999), 291-300, math.CO/9806076.
  21. Chan C.S., Robbins D.P., Yuen D.S., On the volume of a certain polytope, Experiment. Math. 9 (2000), 91-99, math.CO/9810154.
  22. Chaumont L., Liu R., Coding multitype forests: application to the law of the total progeny of branching forests and to enumeration, arXiv:1302.0195.
  23. Chen W.Y.C., A general bijective algorithm for trees, Proc. Nat. Acad. Sci. USA 87 (1990), 9635-9639.
  24. Cherednik I., Double affine Hecke algebras, London Mathematical Society Lecture Note Series, Vol. 319, Cambridge University Press, Cambridge, 2005.
  25. Chervov A., Falqui G., Manin matrices and Talalaev's formula, J. Phys. A: Math. Theor. 41 (2008), 194006, 28 pages, arXiv:0711.2236.
  26. Clark E., Ehrenborg R., Explicit expressions for the extremal excedance set statistics, European J. Combin. 31 (2010), 270-279.
  27. Cohen F.R., Pakianathan J., Vershinin V.V., Wu J., Basis-conjugating automorphisms of a free group and associated Lie algebras, in Groups, Homotopy and Configuration Spaces, Geom. Topol. Monogr., Vol. 13, Geom. Topol. Publ., Coventry, 2008, 147-168, math.GR/0610946.
  28. Cordovil R., A commutative algebra for oriented matroids, Discrete Comput. Geom. 27 (2002), 73-84.
  29. Deift P., Li L.C., Nanda T., Tomei C., The Toda flow on a generic orbit is integrable, Comm. Pure Appl. Math. 39 (1986), 183-232.
  30. Deligne P., Les immeubles des groupes de tresses généralisés, Invent. Math. 17 (1972), 273-302.
  31. Deutsch E., Sagan B.E., Congruences for Catalan and Motzkin numbers and related sequences, J. Number Theory 117 (2006), 191-215, math.CO/0407326.
  32. Di Francesco P., A refined Razumov-Stroganov conjecture, J. Stat. Mech. Theory Exp. 2004 (2004), P08009, 16 pages, cond-mat/0407477.
  33. Di Francesco P., Zinn-Justin P., The quantum Knizhnik-Zamolodchikov equation, generalized Razumov-Stroganov sum rules and extended Joseph polynomials, J. Phys. A: Math. Gen. 38 (2005), L815-L822, math-ph/0508059.
  34. Di Francesco P., Zinn-Justin P., Inhomogeneous model of crossing loops and multidegrees of some algebraic varieties, Comm. Math. Phys. 262 (2006), 459-487, math-ph/0412031.
  35. Dunkl C.F., Differential-difference operators associated to reflection groups, Trans. Amer. Math. Soc. 311 (1989), 167-183.
  36. Dunkl C.F., Harmonic polynomials and peak sets of reflection groups, Geom. Dedicata 32 (1989), 157-171.
  37. Eğecioğlu Ö., Redmond T., Ryavec C., From a polynomial Riemann hypothesis to alternating sign matrices, Electron. J. Combin. 8 (2001), no. 1, 36, 51 pages.
  38. Erdélyi A., Etherington I.M.H., Some problems of non-associative combinatorics (2), Edinburgh Math. Notes 32 (1940), 7-14.
  39. Escobar L., Mészáros K., Subword complexes via triangulations of root polytopes, arXiv:1502.03997.
  40. Felder G., Pasquier V., A simple construction of elliptic $R$-matrices, Lett. Math. Phys. 32 (1994), 167-171, hep-th/9402011.
  41. Fomin S., Gelfand S., Postnikov A., Quantum Schubert polynomials, J. Amer. Math. Soc. 10 (1997), 565-596.
  42. Fomin S., Kirillov A.N., Yang-Baxter equation, symmetric functions and Grothendieck polynomials, hep-th/9306005.
  43. Fomin S., Kirillov A.N., The Yang-Baxter equation, symmetric functions, and Schubert polynomials, Discrete Math. 153 (1996), 123-143.
  44. Fomin S., Kirillov A.N., Reduced words and plane partitions, J. Algebraic Combin. 6 (1997), 311-319.
  45. Fomin S., Kirillov A.N., Quadratic algebras, Dunkl elements, and Schubert calculus, in Advances in Geometry, Progr. Math., Vol. 172, Birkhäuser Boston, Boston, MA, 1999, 147-182.
  46. Fomin S., Procesi C., Fibered quadratic Hopf algebras related to Schubert calculus, J. Algebra 230 (2000), 174-183.
  47. Fulton W., Universal Schubert polynomials, Duke Math. J. 96 (1999), 575-594, alg-geom/9702012.
  48. Ganter N., Ram A., Generalized Schubert calculus, J. Ramanujan Math. Soc. 28A (2013), 149-190, arXiv:1212.5742.
  49. Gel'fand I.M., Rybnikov G.L., Algebraic and topological invariants of oriented matroids, Soviet Math. Dokl. 40 (1990), 148-152.
  50. Gessel I.M., A combinatorial proof of the multivariable Lagrange inversion formula, J. Combin. Theory Ser. A 45 (1987), 178-195.
  51. Gessel I.M., Sagan B.E., The Tutte polynomial of a graph, depth-first search, and simplicial complex partitions, Electron. J. Combin. 3 (1996), no. 2, 9, 36 pages.
  52. Ginzburg V., Kapranov M., Vasserot E., Elliptic algebras and equivariant elliptic cohomology, q-alg/9505012.
  53. Givental A., Stationary phase integrals, quantum Toda lattices, flag manifolds and the mirror conjecture, in Topics in singularity theory, Amer. Math. Soc. Transl. Ser. 2, Vol. 180, Amer. Math. Soc., Providence, RI, 1997, 103-115, alg-geom/9612001.
  54. Gorbounov V., Rimányi R., Tarasov V., Varchenko A., Quantum cohomology of the cotangent bundle of a flag variety as a Yangian Bethe algebra, J. Geom. Phys. 74 (2013), 56-86, arXiv:1204.5138.
  55. Haiman M.D., Conjectures on the quotient ring by diagonal invariants, J. Algebraic Combin. 3 (1994), 17-76.
  56. Hikami K., Wadati M., Topics in quantum integrable systems, J. Math. Phys. 44 (2003), 3569-3594.
  57. Hivert F., Novelli J.-C., Thibon J.-Y., Commutative combinatorial Hopf algebras, J. Algebraic Combin. 28 (2008), 65-95, math.CO/0605262.
  58. Hurtado F., Noy M., Graph of triangulations of a convex polygon and tree of triangulations, Comput. Geom. 13 (1999), 179-188.
  59. Isaev A.P., Kirillov A.N., Bethe subalgebras in Hecke algebra and Gaudin models, Lett. Math. Phys. 104 (2014), 179-193, arXiv:1302.6495.
  60. Isaev A.P., Kirillov A.N., Tarasov V.O., Bethe subalgebras in affine Birman-Murakami-Wenzl algebras and flat connections for $q$-KZ equations, arXiv:1510.05374.
  61. Jensen C., McCammond J., Meier J., The integral cohomology of the group of loops, Geom. Topol. 10 (2006), 759-784, arXiv:0903.0140.
  62. Jucys A.-A.A., Symmetric polynomials and the center of the symmetric group ring, Rep. Math. Phys. 5 (1974), 107-112.
  63. Kajihara Y., Noumi M., Multiple elliptic hypergeometric series. An approach from the Cauchy determinant, Indag. Math. (N.S.) 14 (2003), 395-421, math.CA/0306219.
  64. Kaneko M., Poly-Bernoulli numbers, J. Théor. Nombres Bordeaux 9 (1997), 221-228.
  65. Kawahara Y., On matroids and Orlik-Solomon algebras, Ann. Comb. 8 (2004), 63-80.
  66. Kirillov A.N., On some quadratic algebras, in L.D. Faddeev's Seminar on Mathematical Physics, Amer. Math. Soc. Transl. Ser. 2, Vol. 201, Amer. Math. Soc., Providence, RI, 2000, 91-113, q-alg/9705003.
  67. Kirillov A.N., On some quadratic algebras: Jucys-Murphy and Dunkl elements, in Calogero-Moser-Sutherland Models (Montréal, QC, 1997), CRM Ser. Math. Phys., Springer, New York, 2000, 231-248.
  68. Kirillov A.N., $t$-deformations of quantum Schubert polynomials, Funkcial. Ekvac. 43 (2000), 57-69, math.QA/9802001.
  69. Kirillov A.N., Ubiquity of Kostka polynomials, in Physics and Combinatorics 1999 (Nagoya), World Sci. Publ., River Edge, NJ, 2001, 85-200, math.QA/9912094.
  70. Kirillov A.N., On some algebraic and combinatorial properties of Dunkl elements, Internat. J. Modern Phys. B 26 (2012), 1243012, 28 pages.
  71. Kirillov A.N., Notes on Schubert, Grothendieck and Key polynomials, arXiv:1501.07337.
  72. Kirillov A.N., On some quadratic algebras. II, unpublished.
  73. Kirillov A.N., Maeno T., Noncommutative algebras related with Schubert calculus on Coxeter groups, European J. Combin. 25 (2004), 1301-1325, math.CO/0310068.
  74. Kirillov A.N., Maeno T., Braided differential structure on Weyl groups, quadratic algebras, and elliptic functions, Int. Math. Res. Not. 2008 (2008), no. 14, rnn046, 23 pages, arXiv:0709.4599.
  75. Kirillov A.N., Maeno T., Extended quadratic algebra and a model of the equivariant cohomology ring of flag varieties, St. Petersburg Math. J. 22 (2011), 447-462, arXiv:0712.2580.
  76. Kirillov A.N., Maeno T., A note on quantum $K$-theory of flag varieties, in preparation.
  77. Knutson A., Miller E., Gröbner geometry of Schubert polynomials, Ann. of Math. 161 (2005), 1245-1318, math.AG/0110058.
  78. Knutson A., Miller E., Yong A., Gröbner geometry of vertex decompositions and of flagged tableaux, J. Reine Angew. Math. 630 (2009), 1-31, math.AG/0502144.
  79. Koornwinder T.H., On the equivalence of two fundamental theta identities, Anal. Appl. (Singap.) 12 (2014), 711-725, arXiv:1401.5368.
  80. Kostant B., The solution to a generalized Toda lattice and representation theory, Adv. Math. 34 (1979), 195-338.
  81. Krattenthaler C., Determinants of (generalised) Catalan numbers, J. Statist. Plann. Inference 140 (2010), 2260-2270, arXiv:0709.3044.
  82. Kreweras G., Une famille de polynômes ayant plusieurs propriétés énumeratives, Period. Math. Hungar. 11 (1980), 309-320.
  83. Kuperberg G., Symmetry classes of alternating-sign matrices under one roof, Ann. of Math. 156 (2002), 835-866, math.CO/0008184.
  84. Lascoux A., Anneau de Grothendieck de la variété de drapeaux, in The Grothendieck Festschrift, Vol. III, Progr. Math., Vol. 88, Birkhäuser Boston, Boston, MA, 1990, 1-34.
  85. Lascoux A., Leclerc B., Thibon J.-Y., Flag varieties and the Yang-Baxter equation, Lett. Math. Phys. 40 (1997), 75-90, q-alg/9607015.
  86. Lascoux A., Schützenberger M.P., Symmetry and flag manifolds, in Invariant Theory (Montecatini, 1982), Lecture Notes in Math., Vol. 996, Springer, Berlin, 1983, 118-144.
  87. Lascoux A., Schützenberger M.P., Symmetrization operators in polynomial rings, Funct. Anal. Appl. 21 (1987), 324-326.
  88. Lazard M., Sur les groupes de Lie formels à un paramètre, Bull. Soc. Math. France 83 (1955), 251-274.
  89. Liu R.I., On the commutative quotient of Fomin-Kirillov algebras, arXiv:1409.4872.
  90. Loday J.-L., Inversion of integral series enumerating planar trees, Sém. Lothar. Combin. 53 (2005), B53d, 16 pages, math.CO/0403316.
  91. Loday J.-L., The multiple facets of the associahedron, available at http://www-irma.u-strasbg.fr/~loday/PAPERS/MultFAsENG2.pdf.
  92. Macdonald I.G., Notes on Schubert polynomials, Publications du LaCIM, Vol. 6, Université du Québec à Montréal, 1991.
  93. Macdonald I.G., Symmetric functions and Hall polynomials, 2nd ed., Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1995.
  94. Manin Yu.I., Some remarks on Koszul algebras and quantum groups, Ann. Inst. Fourier (Grenoble) 37 (1987), 191-205.
  95. Manin Yu.I., Quantum groups and noncommutative geometry, Université de Montréal, Centre de Recherches Mathématiques, Montreal, QC, 1988.
  96. Mathieu O., The symplectic operad, in Functional Analysis on the Eve of the 21st Century, Vol. 1 (New Brunswick, NJ, 1993), Progr. Math., Vol. 131, Birkhäuser Boston, Boston, MA, 1995, 223-243.
  97. Matsumoto S., Novak J., Unitary matrix integrals, primitive factorizations, and Jucys-Murphy elements, in 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010), Discrete Math. Theor. Comput. Sci. Proc., AN, Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 2010, 403-411, arXiv:1005.0151.
  98. Merino C., Ramírez-Ibáñez M., Rodríguez-Sánchez G., The Tutte polynomial of some matroids, Int. J. Comb. 2012 (2012), 430859, 40 pages, arXiv:1203.0090.
  99. Mészáros K., Root polytopes, triangulations, and the subdivision algebra. I, Trans. Amer. Math. Soc. 363 (2011), 4359-4382, arXiv:0904.2194.
  100. Mészáros K., Root polytopes, triangulations, and the subdivision algebra. II, Trans. Amer. Math. Soc. 363 (2011), 6111-6141, arXiv:0904.3339.
  101. Mészáros K., Product formula for volumes of flow polytopes, 2015, Proc. Amer. Math. Soc. 143 (2015), 937-954, arXiv:1111.5634.
  102. Mészáros K., $h$-polynomials via reduced forms, arXiv:1407.2685.
  103. Mészáros K., Morales A.H., Flow polytopes of signed graphs and the Kostant partition function, Int. Math. Res. Not. 2015 (2015), 830-871, arXiv:1208.0140.
  104. Mészáros K., Panova G., Postnikov A., Schur times Schubert via the Fomin-Kirillov algebra, Electron. J. Combin. 21 (2014), 1.39, 22 pages, arXiv:1210.1295.
  105. Miller E., Sturmfels B., Combinatorial commutative algebra, Graduate Texts in Mathematics, Vol. 227, Springer-Verlag, New York, 2005.
  106. Molev A., Yangians and classical Lie algebras, Mathematical Surveys and Monographs, Vol. 143, Amer. Math. Soc., Providence, RI, 2007.
  107. Motegi K., Sakai K., Vertex models, TASEP and Grothendieck polynomials, J. Phys. A: Math. Theor. 46 (2013), 355201, 26 pages, arXiv:1305.3030.
  108. Mukhin E., Tarasov V., Varchenko A., Bethe subalgebras of the group algebra of the symmetric group, Transform. Groups 18 (2013), 767-801, arXiv:1004.4248.
  109. Mutafyan G.S., Feigin B.L., Characters of representations of the quantum toroidal algebra $\widehat{\widehat{\mathfrak{gl}}}_1$: plane partitions with ''stands'', Funct. Anal. Appl. 48 (2014), 36-48.
  110. Novelli J.-C., Thibon J.-Y., Hopf algebras of $m$-permutations, $(m+1)$-ary trees, and $m$-parking functions, arXiv:1403.5962.
  111. Okada S., Enumeration of symmetry classes of alternating sign matrices and characters of classical groups, J. Algebraic Combin. 23 (2006), 43-69, math.CO/0408234.
  112. Orellana R., Ram A., Affine braids, Markov traces and the category ${\mathcal O}$, in Algebraic Groups and Homogeneous Spaces, Tata Inst. Fund. Res. Stud. Math., Tata Inst. Fund. Res., Mumbai, 2007, 423-473, math.RT/0401317.
  113. Orlik P., Terao H., Arrangements of hyperplanes, Grundlehren der Mathematischen Wissenschaften, Vol. 300, Springer-Verlag, Berlin, 1992.
  114. Orlik P., Terao H., Commutative algebras for arrangements, Nagoya Math. J. 134 (1994), 65-73.
  115. Polishchuk A., Classical Yang-Baxter equation and the $A_\infty$-constraint, Adv. Math. 168 (2002), 56-95, math.AG/0008156.
  116. Polishchuk A., Positselski L., Quadratic algebras, University Lecture Series, Vol. 37, Amer. Math. Soc., Providence, RI, 2005.
  117. Postnikov A., On a quantum version of Pieri's formula, in Advances in geometry, Progr. Math., Vol. 172, Birkhäuser Boston, Boston, MA, 1999, 371-383.
  118. Postnikov A., Shapiro B., Shapiro M., Algebras of curvature forms on homogeneous manifolds, in Differential Topology, Infinite-Dimensional Lie algebras, and Applications, Amer. Math. Soc. Transl. Ser. 2, Vol. 194, Amer. Math. Soc., Providence, RI, 1999, 227-235, math.AG/9901075.
  119. Postnikov A., Stanley R.P., Deformations of Coxeter hyperplane arrangements, J. Combin. Theory Ser. A 91 (2000), 544-597, math.CO/9712213.
  120. Proctor R.A., Bruhat lattices, plane partition generating functions, and minuscule representations, European J. Combin. 5 (1984), 331-350.
  121. Proctor R.A., New symmetric plane partition identities from invariant theory work of De Concini and Procesi, European J. Combin. 11 (1990), 289-300.
  122. Proctor R.A., Product evaluations of Lefschetz determinants for Grassmannians and of determinants of multinomial coefficients, J. Combin. Theory Ser. A 54 (1990), 235-247.
  123. Pyatov P., Raise and peel models of fluctuating interfaces and combinatorics of Pascal's hexagon, J. Stat. Mech. Theory Exp. 2004 (2004), P09003, 30 pages, math-ph/0406025.
  124. Ryom-Hansen S., On the representation theory of an algebra of braids and ties, J. Algebraic Combin. 33 (2011), 57-79, arXiv:0801.3633.
  125. Saito K., Growth function of Artin monoids, Proc. Japan Acad. Ser. A Math. Sci. 85 (2009), no. 7, 84-88.
  126. Schenck H., Tohaneanu Ş.O., The Orlik-Terao algebra and 2-formality, Math. Res. Lett. 16 (2009), 171-182, arXiv:0901.0253.
  127. Schuetz A., Whieldon G., Polygonal dissections and reversions of series, arXiv:1401.7194.
  128. Serrano L., Stump C., Generalized triangulations, pipe dreams, and simplicial spheres, in 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011), Discrete Math. Theor. Comput. Sci. Proc., AO, Assoc. Discrete Math. Theor. Comput. Sci., Nancy, 2011, 885-896.
  129. Shapiro B., Shapiro M., On ring generated by Chern $2$-forms on ${\rm SL}_n/B$, C. R. Acad. Sci. Paris Sér. I Math. 326 (1998), 75-80.
  130. Shibukawa Y., Ueno K., Completely ${\bf Z}$ symmetric $R$ matrix, Lett. Math. Phys. 25 (1992), 239-248.
  131. Sloan N., The on-line encyclopedia of integer sequences, available at https://oeis.org/.
  132. Stanley R.P., Acyclic flow polytopes and Kostant's partition function, Conference transparencies, 2000, available at http://www-math.mit.edu/~rstan/transparencies/kostant.ps.
  133. Stanley R.P., Catalan addendum, version of April 30, 2011, available at htpp:/www-math.mit.edu/~rstan/ec/catadd.pdf.
  134. Stanley R.P., Enumerative combinatorics. Vol. 2, Cambridge Studies in Advanced Mathematics, Vol. 62, Cambridge University Press, Cambridge, 1999.
  135. Stump C., A new perspective on $k$-triangulations, J. Combin. Theory Ser. A 118 (2011), 1794-1800, arXiv:1009.4101.
  136. Sulanke R.A., Counting lattice paths by Narayana polynomials, Electron. J. Combin. 7 (2000), 40, 9 pages.
  137. Tamvakis H., Arithmetic intersection theory on flag varieties, Math. Ann. 314 (1999), 641-665, alg-geom/9611006.
  138. Uglov D., Finite-difference representations of the degenerate affine Hecke algebra, Phys. Lett. A 199 (1995), 353-359, hep-th/9409155.
  139. Wachs M.L., Flagged Schur functions, Schubert polynomials, and symmetrizing operators, J. Combin. Theory Ser. A 40 (1985), 276-289.
  140. Welsh D., The Tutte polynomial, Random Structures Algorithms 15 (1999), 210-228.
  141. Whittaker E.T., Watson G.N., A course of modern analysis, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1996.
  142. Woo A., Catalan numbers and Schubert polynomials for $w=1(n+1)\cdots 2$, math.CO/0407160.

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