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{"Title": "Augmented Chow polynomials of matroids", "Definition": "For a finite simple matroid $M$, the augmented Chow polynomial $H_M(t)=\\sum_{i\\geq0}\\dim_{\\mathbb Q}\\mathrm{CH}^i(M)t^i$ is the Hilbert\u2013Poincar\u00e9 series of its augmented Chow ring CITE{FMSV}, Definition 3.2. Every divisor generator has degree one.", "Parameters": {"matroid": {"type": "Symbolic", "title": "matroid", "display": "$M$", "constraints": "a finite simple matroid", "values": {"U(3,4)": "$U_{3,4}$", "U(3,5)": "$U_{3,5}$", "K(4)": "$M(K_{4})$", "U(3,6)": "$U_{3,6}$", "Fano": "Fano", "NonFano": "NonFano", "U(3,7)": "$U_{3,7}$", "NonPappus": "NonPappus", "Pappus": "Pappus", "U(4,5)": "$U_{4,5}$", "U(4,6)": "$U_{4,6}$", "U(4,7)": "$U_{4,7}$", "U(4,8)": "$U_{4,8}$", "Vamos": "V\u00e1mos", "K(5)": "$M(K_{5})$", "U(5,6)": "$U_{5,6}$", "U(5,7)": "$U_{5,7}$", "U(5,8)": "$U_{5,8}$", "TicTacToe": "TicTacToe", "U(5,9)": "$U_{5,9}$", "R10": "$R_{10}$", "K(6)": "$M(K_{6})$", "U(6,7)": "$U_{6,7}$", "U(6,8)": "$U_{6,8}$", "U(6,9)": "$U_{6,9}$", "K5dual": "$M^*(K_5)$", "U(6,10)": "$U_{6,10}$", "K(7)": "$M(K_{7})$", "U(7,8)": "$U_{7,8}$", "U(7,9)": "$U_{7,9}$", "U(7,10)": "$U_{7,10}$", "U(7,11)": "$U_{7,11}$", "K(8)": "$M(K_{8})$", "U(8,9)": "$U_{8,9}$", "U(8,10)": "$U_{8,10}$", "U(8,11)": "$U_{8,11}$", "U(8,12)": "$U_{8,12}$"}}}, "Comments": {"comment-ring": "Write $E$ for the ground set and $L(M)$ for the lattice of flats. The augmented ring is generated over $\\mathbb Q$ by $x_F$ for all proper flats $F$, including $\\varnothing$, and by $y_e$ for $e\\in E$. Its relations are $y_e=\\sum_{F\\not\\ni e}x_F$, $x_Fx_G=0$ for incomparable flats, and $y_ex_F=0$ when $e\\notin F$ CITE{FMSV}, Definition 3.2. This is distinct from the HREF{Chow_polynomials_of_matroids}[ordinary Chow ring] with the maximal building set of all nonempty flats; a minimal building set gives a different construction.", "comment-notation": "Let $r=\\operatorname{rk}M$ and let $\\mu$ be the M\u00f6bius function of $L(M)$. Contraction $M/F$ corresponds to the upper interval $[F,E]$, whereas restriction $M|F$ corresponds to $[\\varnothing,F]$. Empty minors have $H=1$. Contractions can have parallel elements, and their actual flat lattices determine the values. The parameter domain excludes looped inputs.", "comment-objects": "The key $U(r,n)$ denotes the uniform matroid $U_{r,n}$ of rank $r$ on $n$ elements. The key $K(m)$ denotes the braid matroid $M(K_m)$, the cycle matroid of the complete graph on $m$ vertices, of rank $m-1$ and size $\\binom m2$; its flat lattice is the partition lattice CITE{FMSV}, Section 5.2. Fano, NonFano, Pappus, NonPappus, Vamos, R10, TicTacToe and K5dual are the exact Sage catalogue objects CITE{SageObjects}; K5dual means $M^*(K_5)$.", "comment-grading": "The variable $t$ records algebraic degree, so cohomological degree $2i$ contributes $t^i$ in a geometric realization. The coefficients are nonnegative, palindromic and unimodal CITE{FMSV}, Section 3.1."}, "Formulas": {"formula-chow": "With $\\underline H_N(t)$ denoting the HREF{Chow_polynomials_of_matroids}[ordinary Chow polynomial], $H_M(t)=\\sum_{F\\in L(M)}t^{\\operatorname{rk}F}\\underline H_{M/F}(t)$ CITE{FMSV}, Proposition 3.7. Equivalently, $H_M(t)=1+t\\sum_{F\\ne E}\\underline H_{M/F}(t)$, by Remark 3.8.", "formula-mobius": "For positive rank, $H_M(t)=-\\sum_{\\varnothing\\ne F\\in L(M)}\\mu(\\varnothing,F)(1+t+\\cdots+t^{\\operatorname{rk}F})H_{M/F}(t)$ CITE{FMSV}, Theorem 3.14.", "formula-uniform": "$H_{U_{r,n}}(t)=1+t\\sum_{j=0}^{r-1}\\binom nj A_j(t)(1+t+\\cdots+t^{r-1-j})$, where $A_j(t)=\\sum_{\\sigma\\in S_j}t^{\\operatorname{des}\\sigma}$, $\\operatorname{des}\\sigma$ counts descents and $A_0(t)=1$ CITE{FMSV}, Theorem 3.21.", "formula-symmetry": "$\\deg H_M=r$, $H_M(0)=1$ and $t^rH_M(t^{-1})=H_M(t)$ CITE{FMSV}, Section 3.1.", "formula-linear": "$[t]H_M(t)=|L(M)|-1$. Eliminating the variables $y_e$ in the augmented ring presentation leaves one independent degree-one generator for each proper flat."}, "Programs": {"program-sage": {"language": "Sage", "code": "# Place the attached generate.py in the working directory.\nfrom generate import MatroidAugmentedChow, uniform_augmented, braid_augmented\nuniform_augmented(8, 12)\nbraid_augmented(8)\n\ngenerator = MatroidAugmentedChow()\n[(p[\"matroid\"], generator.value(p, 100)) for p in generator.enumerate()]"}}, "Similar tables": [{"table": "HREF{Chow_polynomials_of_matroids}[Chow polynomials of matroids]", "relation": "holds the ordinary Chow polynomials $\\underline H_M(t)$, which determine $H_M(t)$ through their values on contractions in CITE{formula-chow}"}, {"table": "HREF{Z-polynomials_of_matroids}[Z-polynomials of matroids]", "relation": "holds the Hilbert series of the matroid intersection cohomology modules, indexed by the same matroids"}, {"table": "HREF{Characteristic_polynomials_of_matroids}[Characteristic polynomials of matroids]", "relation": "holds $\\chi_M(t)$, whose constant term is $\\mu(\\varnothing,E)$; restrictions supply the M\u00f6bius coefficients in CITE{formula-mobius}"}], "Links": {"SageObjects": {"title": "SageMath: database of matroids", "url": "https://doc.sagemath.org/html/en/reference/matroids/sage/matroids/database_matroids.html"}}, "References": {"FMSV": {"bib": "Luis Ferroni, Jacob P. Matherne, Matthew Stevens and Lorenzo Vecchi, Hilbert\u2013Poincar\u00e9 series of matroid Chow rings and intersection cohomology, Advances in Mathematics 449 (2024), 109733.", "arxiv": "2212.03190", "doi": "10.1016/j.aim.2024.109733"}}, "Keywords": ["augmented matroid Chow ring", "Hilbert\u2013Poincar\u00e9 series", "Hilbert-Poincare series", "uniform matroid", "braid matroid", "partition lattice", "Fano", "NonFano", "Pappus", "NonPappus", "Vamos", "V\u00e1mos", "R10", "TicTacToe", "K5dual"], "Tags": ["polynomial", "combinatorics", "algebra"], "Data properties": {"type": "Z[]", "rigour": "exact", "complete": "no", "complete-note": "it holds all 24 uniform matroids $U_{r,r+c}$ with $3\\leq r\\leq8$ and $1\\leq c\\leq4$, all five braid matroids $M(K_m)$ with $4\\leq m\\leq8$, and the eight Sage catalogue objects Fano, NonFano, Pappus, NonPappus, Vamos, R10, TicTacToe and K5dual. The four low-corank uniform slices include circuits and compare generic dependence across ranks; the braid family contributes contrasting partition lattices. The eight named objects are a noncanonical exploratory reference selection spanning finite geometry, representability contrasts, regular, paving and cographic structures. The selection is independent of numeral base and ground-set labels. These practical bounds favor breadth across these families before further depth; the named objects have at most ten elements and $K(8)$ has $4140$ flats. No completeness over all matroids of any rank or size is claimed", "rigour details": "All 37 values are exact polynomials in $\\mathbb Z[t]$. The generator evaluates the weighted-chain formula of CITE{FMSV}, Proposition 3.6, by dynamic programming on upper flat intervals, with symmetry reductions for uniform and braid matroids.\n\nThe separate checks.py uses CITE{formula-mobius} for the eight catalogue objects, constructing flats independently from all subset ranks, and permutation-cycle counts for the five braid objects. It checks all 24 uniform values using CITE{formula-uniform}, with Eulerian coefficients counted directly from permutations. Ten complete polynomials also agree with published Table 1(b,d). Known-answer controls precede the comparisons. Degree, endpoint coefficients, nonnegativity, symmetry, unimodality and CITE{formula-linear} are checked on every entry, as are both expressions in CITE{formula-chow}. The standard dry run measures a longest entry of 95 characters and about 3.4 KB of entries. The attached argument-selection.md records the coverage rationale, and numerical-evidence.md records the commands and results."}, "Display properties": {"number-header": "$H_M(t)$", "group parameters": [["matroid"]]}, "Numbers": {"U(3,4)": "t^3 + 11*t^2 + 11*t + 1", "U(3,5)": "t^3 + 16*t^2 + 16*t + 1", "K(4)": "t^3 + 14*t^2 + 14*t + 1", "U(3,6)": "t^3 + 22*t^2 + 22*t + 1", "Fano": "t^3 + 15*t^2 + 15*t + 1", "NonFano": "t^3 + 17*t^2 + 17*t + 1", "U(3,7)": "t^3 + 29*t^2 + 29*t + 1", "NonPappus": "t^3 + 30*t^2 + 30*t + 1", "Pappus": "t^3 + 28*t^2 + 28*t + 1", "U(4,5)": "t^4 + 26*t^3 + 66*t^2 + 26*t + 1", "U(4,6)": "t^4 + 42*t^3 + 117*t^2 + 42*t + 1", "U(4,7)": "t^4 + 64*t^3 + 190*t^2 + 64*t + 1", "U(4,8)": "t^4 + 93*t^3 + 289*t^2 + 93*t + 1", "Vamos": "t^4 + 78*t^3 + 234*t^2 + 78*t + 1", "K(5)": "t^4 + 51*t^3 + 146*t^2 + 51*t + 1", "U(5,6)": "t^5 + 57*t^4 + 302*t^3 + 302*t^2 + 57*t + 1", "U(5,7)": "t^5 + 99*t^4 + 610*t^3 + 610*t^2 + 99*t + 1", "U(5,8)": "t^5 + 163*t^4 + 1115*t^3 + 1115*t^2 + 163*t + 1", "TicTacToe": "t^5 + 224*t^4 + 1576*t^3 + 1576*t^2 + 224*t + 1", "U(5,9)": "t^5 + 256*t^4 + 1888*t^3 + 1888*t^2 + 256*t + 1", "R10": "t^5 + 161*t^4 + 986*t^3 + 986*t^2 + 161*t + 1", "K(6)": "t^5 + 202*t^4 + 1412*t^3 + 1412*t^2 + 202*t + 1", "U(6,7)": "t^6 + 120*t^5 + 1191*t^4 + 2416*t^3 + 1191*t^2 + 120*t + 1", "U(6,8)": "t^6 + 219*t^5 + 2641*t^4 + 5637*t^3 + 2641*t^2 + 219*t + 1", "U(6,9)": "t^6 + 382*t^5 + 5290*t^4 + 11674*t^3 + 5290*t^2 + 382*t + 1", "K5dual": "t^6 + 313*t^5 + 3638*t^4 + 7628*t^3 + 3638*t^2 + 313*t + 1", "U(6,10)": "t^6 + 638*t^5 + 9773*t^4 + 22073*t^3 + 9773*t^2 + 638*t + 1", "K(7)": "t^6 + 876*t^5 + 13623*t^4 + 31018*t^3 + 13623*t^2 + 876*t + 1", "U(7,8)": "t^7 + 247*t^6 + 4293*t^5 + 15619*t^4 + 15619*t^3 + 4293*t^2 + 247*t + 1", "U(7,9)": "t^7 + 466*t^6 + 10204*t^5 + 40444*t^4 + 40444*t^3 + 10204*t^2 + 466*t + 1", "U(7,10)": "t^7 + 848*t^6 + 21995*t^5 + 92255*t^4 + 92255*t^3 + 21995*t^2 + 848*t + 1", "U(7,11)": "t^7 + 1486*t^6 + 43715*t^5 + 190730*t^4 + 190730*t^3 + 43715*t^2 + 1486*t + 1", "K(8)": "t^7 + 4139*t^6 + 135417*t^5 + 618242*t^4 + 618242*t^3 + 135417*t^2 + 4139*t + 1", "U(8,9)": "t^8 + 502*t^7 + 14608*t^6 + 88234*t^5 + 156190*t^4 + 88234*t^3 + 14608*t^2 + 502*t + 1", "U(8,10)": "t^8 + 968*t^7 + 36605*t^6 + 247397*t^5 + 452357*t^4 + 247397*t^3 + 36605*t^2 + 968*t + 1", "U(8,11)": "t^8 + 1816*t^7 + 83777*t^6 + 610556*t^5 + 1139876*t^4 + 610556*t^3 + 83777*t^2 + 1816*t + 1", "U(8,12)": "t^8 + 3302*t^7 + 177201*t^6 + 1361494*t^5 + 2578369*t^4 + 1361494*t^3 + 177201*t^2 + 3302*t + 1"}}