Since in $D$ the $\mathbb{L}$ in column $\ell_{s+1}$ glues to the left of the $\mathbb{L}$ in column $\ell_{s}$ at its bottom-most $\leplus$, there must be $0$'s in row $b_{s+1}$ in-between columns $\ell_s$ and $\ell_{s+1}$.
Therefore, we get the above figure on the right.

For column $j=\ell_{m+1}$, notice that in $\hat{D}$, there must be a row $b < b_m$ with a $\leplus$ to the left of column $j$ since we know at the least there's a $\leplus$ at $(1, a_n)$.
Since we know that $(b_m,j)$ is the first $\leplus$ in column $j$ and that $b_T \neq j, a_n$ for the $\mathbb{L}$ in column $j$ in $D$, we must have that $b_T = b$ and thus there is a $\leplus$ at $(b, j)$ with 0's all above in $D$.
For column $\ell_m$, using the same argument as for $\ell_s$ with row $b$ as $b_{s+1}$, in the $\mathbb{L}$ in column $\ell_m$, there must be a $\leplus$ at $(b, \ell_m)$ and at $(b_m, \ell_m)$ with 0's in-between.

Since we know that $\mathbb{L}$'s glue to the left of previous $\mathbb{L}$'s at the first $\leplus$ from the bottom, we get the following in $D$, regardless if $i+1$ is a column or a row:

\[ \permproofi \]

Finally to look at what happens with $i$ in $D$.

* If $i+1 = \ell_1$ and $i$ are both columns, then regardless of how many $\leplus$'s there are in column $i$ in $\hat{D}$, we must have in $D$ a $\leplus$ at $(b_1, i)$ and 0's below in column $i-1$ since we know how $\mathbb{L}$'s glue.

\[ \permproofibothcols \]

* If $i+1 = \ell_1$ and $i$ is a row, then we must have $b_1 = i$ (since $\hat{D}$ is co-loopless).

\[ \permprooficolrow \]

* If $i+1$ is a row and $i$ is a row, then similarly we must have $b_1 = i$ (since $\hat{D}$ is co-loopless and by the -condition as there are $\leplus$'s at $(i+1,\ell_1), (b_1,\ell_1)$).

\[ \permproofibothrows \]

* If $i+1$ is a row and $i$ is a column, then notice $b_1 < i$ and is a row in $\hat{D}$ with its last $\leplus$ to the left of column $i$.
If $i$ is a column of all 0's in $\hat{D}$, then Step 2(iii) for row $b_1$ would place the only $\leplus$ in column $i$ in $D$ at $(b_1, i)$.
Otherwise, we can use the same argument for $i$ as for $\ell_s$, taking $\ell_1$ as $\ell_{s+1}$ (and only looking at rows above $i$), to show that there must be a $\leplus$ at $(b_1, i)$ and 0's all below in $D$.

\[ \permproofirowcol \]

In all the cases, we can connect these path starts at the circled point to the rest of the path, and thus we get that the corresponding path to $i$ in $D$ leads to $j$, giving us $\pi(i) = j$ as needed for paths in (i).

For paths in (iv), the proof is essentially the same as for (i) except now we need to consider when $j$ is a row.
The other difference is we no longer have the fact that $b_T \neq f, a_n$ for columns $\ell_s$, which was only used when $(b_s, \ell_s)$ was the first $\leplus$ in its column in $\hat{D}$.
However for paths in (iv), we have that if $(b_s, \ell_s)$ is the first $\leplus$ in the column, then $b_{s+1}$ is the first row above $b_s$ and below $a_n$ with a $\leplus$ to the left of $\ell_s$ (since $a_n < j$) and so we still have $b_T = b_{s+1}$ for the $\mathbb{L}$ in column $\ell_s$.

When $j$ is a row, we have an additional $\leplus$ at $(j, \ell_{m+1})$ in column $\ell_{m+1}$ above $b_m$ in $\hat{D}$.
Now we have $b = j$ for column $\ell_{m}$, and looking at column $\ell_{m+1}$, we notice that the $\leplus$ at $(j, \ell_{m+1})$ is the last $\leplus$ in row $j$.
We also know that row $j$ is of type (I) since $j > a_n$, and thus in the $\mathbb{L}$ in column $\ell_{m+1}$ in $D$, there must be a 0 at $(j, \ell_{m+1})$.

Thus we have, where the dotted line refers to where row $a_n$ will be in $D$:

\[ \permproofiv \]

The rest of the analysis is exactly the same for $i+1$ and $i$, and thus we also get that $\pi(i) = j$ as needed for paths in (iv).

For paths in (ii), first notice we already have the case when the last $\leplus$ in the path is before $a_n$ and $j$ is a row.
In this case, all the arguments from paths in (i) stay the same, and we can use the same argument for column $\ell_{m+1}$ as paths in (iv), since if $(j, \ell_{m+1})$ is the last $\leplus$ in row $j$ with $\ell_{m+1} < a_n$, then row $j$ is also of type (I).

When the path contains $\leplus$'s in column $a_n$, we have (since $j \neq a_n$) either:

\[ \permproofiihat \]

where the dots are indicating $\leplus$'s in column $a_n$ that are not part of the path.

Then notice for columns $\ell_1, \ldots, \ell_{s-1}$, the analysis from paths in (i) stays exactly the same.
For columns $\ell_{s+1}, \ldots, \ell_{m+1}$, we note that all rows $b_{s}, \ldots, b_m$, and $j$ if it is a row, must be of type (II) since by -condition, they must have $\leplus$'s in column $a_n$ as there is a $\leplus$ at $(1, a_n)$.
In particular, this means that the $\mathbb{L}$'s corresponding to columns $\ell_{s+1}, \ldots, \ell_{m+1}$, which are all $> a_n$, has $\leplus$'s exactly in the same positions as in $\hat{D}$.
However now we are done as to get $D$, column $a_n$ is removed and we know how the $\mathbb{L}$'s are glued, so we have:

\[ \permproofii \]

The analysis on $i+1$ and $i$ stays exactly the same as before (since it only depends on $\hat{D}$), even if $i+1 = \ell_1 = a_n$.
In this case, the only difference is that the circled $\leplus$ in the analysis for $i$ in $D$ is now the $\leplus$ at $(b_1, \ell_2)$, and we consider $i+1$ to be a row.
Thus we get that $\pi(i) = j$ as needed for paths in (ii).

Lastly for paths in (iii), where the dotted line refers to where row $a_n$ will be in $D$, we have:

\[ \permproofiiihat \]

Here we essentially have the same proof as for (ii).
Once again for columns $\ell_1, \ldots, \ell_{s-2}$, the analysis from paths in (iv) stays exactly the same.
For columns $\ell_{s+1}, \ldots, \ell_{m+1}$, once again the rows $b_{s}, \ldots, b_{m}, b_{m+1},$ must be of type (II) since they are all $< a_n$ and have $\leplus$'s to the left of column $a_n$, and thus the $\mathbb{L}$'s corresponding to these columns have $\leplus$'s in the exact same positions as in $\hat{D}$.

The only difference is we need to look at what happens to column $\ell_s$, which row $a_n$ will pass through, and column $\ell_{s-1}$.
For column $\ell_s$, since row $a_n$ will occur in-between rows $b_s$ and $b_{s-1}$, there will be a $\leplus$ at $(a_n, \ell_s)$ in the $\mathbb{L}$ corresponding to column $\ell_s$.
Since row $b_s$ is of type (II) with $\ell_s > a_n$, there is also a $\leplus$ at $(b_s, \ell_s)$ in the $\mathbb{L}$.

For column $\ell_{s-1}$, since the $\leplus$ at $(b_{s-1}, \ell_{s-1})$ is not a last $\leplus$ in row $b_{s-1}$, there is a $\leplus$ in the same position in $D$.
Now, if this $\leplus$ at $(b_{s-1}, \ell_{s-1})$ is the first one in column $\ell_{s-1}$ in $\hat{D}$, then we know that the corresponding $\mathbb{L}$ will have its $b_T = a_n$, in which case there is a $\leplus$ at $(a_n, \ell_{s-1})$.
Otherwise if there are $\leplus$'s at $(b_{s-1}, \ell_{s-1})$ in $\hat{D}$, for $b_{s-1} > b > a_n$ these must be last $\leplus$'s in its row with $b$ of type (I) and thus corresponds to a 0 at $(b, \ell_{s-1})$ in $D$.
For $b < a_n$, this means that $a_n$ is in-between the first and last $\leplus$'s in column $\ell_{s-1}$ and thus there must be a $\leplus$ at $(a_n, \ell_{s-1})$.
In either case, the $\mathbb{L}$ in column $\ell_s-1$ always has $\leplus$'s at $(a_n, \ell_{s-1})$ and $(b_{s-1}, \ell_{s-1})$ with 0's in-between and so we have:

\[ \permproofiii \]

In the special case of $a_n < b_m$ where $j = \ell_{m+1} = \ell_s$, then since we know that the $\leplus$ at $(b_m, j)$ is the first in its column in $\hat{D}$, the $\mathbb{L}$ in column $j$ in $D$ has its $b_T = a_n$ with a $\leplus$ at $(a_n, j)$.
Column $\ell_{m} = \ell_{s-1}$ stays the same as before with $\leplus$'s at $(a_n, \ell_m)$ and $(b_m, \ell_m)$, and 0's in-between, in $D$:

\[ \permproofiiispecialone \]

In the special case of $a_n > b_1$ where $i+1 = \ell_1$, now the $\mathbb{L}$ in column $\ell_1 = \ell_s$ in $D$ will have a $\leplus$ at $(a_n, \ell_1)$, since $\ell_1 > a_n$ and thus its $b_B = a_n$.
There is also still a $\leplus$ at $(b_1, \ell_1)$, since $b_1 < a_n$ and thus is of type (II), which aligns with the $\ell_s$ as in the general case.

\[ \permproofiiispecialtwo \]

The analysis on $i+1$ and $i$ stays exactly the same as before, with the slight exception of $a_n > b_1$.
Since we have $i \geq a_n$ with a $\leplus$ at $(a_n, \ell_1)$ in $D$, now $a_n$ plays the role of $b_1$ and we use the same arguments as before, even if $i = a_n$ in which case we consider $i$ to be a row.
In all cases, we get that $\pi(i) = j$ as needed for paths in (iv).

Putting (i)–(iv) together, we are now done since for all paths $\hat\pi(i+1) = j$ we get $\pi(i) = \hat\pi(i+1) = j$!

T-duality on the level of Le diagrams

With the proof that $D$ gives the correct decorated permutation complete, we finally conclude by putting everything together.
Theorems <ref>, <ref> and <ref>, tell us that in fact, the algorithm given in Section <ref> is telling us what the T-duality map (see Section <ref>) looks like on the level of -diagrams.

Given a co-loopless -diagram $\hat{D}$ of type $(k,n)$, the algorithm given in Section <ref> constructs a loopless -diagram $D$ of type $(k+1,n)$ such that

* the dimensions of the positroid cells indexed by $D$ and $\hat{D}$ relate via

\[ \dim(S_D) = \dim(S_{\hat{D}}) - 2k + (n-1), \]

* and the associated decorated permutations to $D$ and $\hat{D}$ relate via T-duality.

That is, the algorithm gives the T-duality map from $\hat{\pi} \mapsto \pi$ on the level of -diagrams.

Final thoughts

CHAPTER: CONCLUSION

CHAPTER: REFERENCES

Missing 'biblatex' package
The bibliography requires the 'biblatex' package.

Cambridge University Press, Cambridge

titleGrassmannian geometry of scattering amplitudes

Springer Science
Business Media LLC

titleThe Amplituhedron
journaltitleJournal of High Energy Physics

titleZeroes of polynomials over finite fields
journaltitleAmer. J. Math.

titleMatroids, motives, and a conjecture of Kontsevich
journaltitleDuke Math. J.

pagesPaper No. 1.13, 20
titleQuotients of uniform positroids
journaltitleElectron. J. Combin.

titleOn motives associated to graph polynomials
journaltitleComm. Math. Phys.

titleKnots and numbers in $\phi^4$ theory to $7$ loops and
journaltitleInternat. J. Modern Phys. C

titleAssociation of multiple zeta values with positive knots via
Feynman diagrams up to $9$ loops
journaltitlePhys. Lett. B

pages181602, 4
titleDirect proof of the tree-level scattering amplitude recursion
relation in Yang-Mills theory
journaltitlePhys. Rev. Lett.

titleThe massless higher-loop two-point function
journaltitleComm. Math. Phys.

titleOn the periods of some Feynman integrals

Cambridge Univ. Press, Cambridge

booktitleGeometric and topological methods for quantum field
titleIterated integrals in quantum field theory

titleA K3 in $\phi^4$
journaltitleDuke Math. J.

titleModular forms in quantum field theory
journaltitleCommun. Number Theory Phys.

titleSingle-valued multiple polylogarithms and a proof of the
zig-zag conjecture
journaltitleJ. Number Theory

titleProperties of $c_2$ invariants of Feynman graphs
journaltitleAdv. Theor. Math. Phys.

titleSpanning forest polynomials and the transcendental weight of
Feynman graphs
journaltitleComm. Math. Phys.

journaltitleAnn. Inst. Henri Poincaré D

titleA combinatorial proof of the all minors matrix tree theorem
journaltitleSIAM J. Algebraic Discrete Methods

titleTableaux combinatorics for the asymmetric exclusion process
journaltitleAdv. in Appl. Math.

title$c_2$ invariants of recursive families of graphs
journaltitleAnn. Inst. Henri Poincaré D

titleOn one example and one counterexample in counting rational
points on graph hypersurfaces
journaltitleLett. Math. Phys.

titleThe $c_2$ invariant is invariant
journaltitleAdv. Theor. Math. Phys.

noteIn preperation
titleA result on the $c_2$ invariant for powers of primes

Cambridge University Press, Cambridge

isbn0-521-56144-2; 0-521-56724-6
noteWith applications to representation theory and geometry
seriesLondon Mathematical Society Student Texts
titleYoung tableaux

titleCombinatorial geometries, convex polyhedra, and Schubert
journaltitleAdv. in Math.

noteAccepted, to appear in Annales de l’Institut Henri
Poincaré D
titleFurther investigations into the graph theory of
$\phi^4$-periods and the $c_2$ invariant

McGraw-Hill International Book Co., New York

seriesInternational Series in Pure and Applied Physics
titleQuantum field theory

titleÜber die Auflösung der Gleichungen, auf welche man
bei der Untersuchung der linearen Vertheilung galvanischer Ströme
geführt wird
journaltitleAnnalen der Physik

titleSchubert calculus
journaltitleAmer. Math. Monthly

Springer, Heidelberg

booktitleAlgebras, quivers and representations
seriesAbel Symp.
titleCombinatorics of KP solitons from the real Grassmannian

journaltitleAnn. Inst. Henri Poincaré D

Springer, Berlin

booktitleMathematics unlimited—2001 and beyond

titleSome results on double triangle descendants of $K_5$
journaltitleAnn. Inst. Henri Poincaré D

titleThe positive tropical Grassmannian, the hypersimplex, and the
m=2 amplituhedron

Springer, Berlin

booktitleTopology and geometry—Rohlin Seminar
seriesLecture Notes in Math.
titleThe universality theorems on the classification problem of
configuration varieties and convex polytopes varieties

Oxford University Press, Oxford

seriesOxford Graduate Texts in Mathematics
titleMatroid theory

titleHepp's bound for Feynman graphs and matroids

titleTotal positivity, Grassmannians, and networks

World Sci. Publ., Hackensack, NJ

booktitleProceedings of the International Congress of
Mathematicians—Rio de Janeiro 2018. Vol. IV. Invited lectures
titlePositive Grassmannian and polyhedral subdivisions

titleThe m=2 amplituhedron and the hypersimplex: signs, clusters,
triangulations, Eulerian numbers

titleMatching polytopes, toric geometry, and the totally
non-negative Grassmannian
journaltitleJ. Algebraic Combin.

family=The Sage Developers,

titleSageMath, the Sage Mathematics Software System
(Version 8.2)

titleQuantum periods: a census of $\phi^4$-transcendentals
journaltitleCommun. Number Theory Phys.

pagesPaper 102, 23
titleQuantum field theory over $\mathbb{F}_q$
journaltitleElectron. J. Combin.

noteWith an appendix by F. Knop
titleGeometries in perturbative quantum field theory
journaltitleCommun. Number Theory Phys.

pagesPaper No. 100, 26
title$c_2$ invariants of hourglass chains via quadratic
denominator reduction
journaltitleSIGMA Symmetry Integrability Geom. Methods Appl.

titleBruhat interval polytopes
journaltitleAdv. Math.

titleA few $c_2$ invariants of circulant graphs
journaltitleCommun. Number Theory Phys.

Springer, Cham

isbn978-3-319-47550-9; 978-3-319-47551-6
seriesSpringerBriefs in Mathematical Physics
titleA combinatorial perspective on quantum field theory

howpublishedPrivate communications

titleA special case of completion invariance for the $c_2$
invariant of a graph
journaltitleCanad. J. Math.

EMS Publ. House, Berlin

booktitleAlgebraic combinatorics, resurgence, moulds and
applications (CARMA). Vol. 2
seriesIRMA Lect. Math. Theor. Phys.
titleA study on prefixes of $c_2$ invariants
year[2020] 2020
[segment=1,heading=subbibliography,title=Chapter 1. Introduction]

Missing 'biblatex' package
The bibliography requires the 'biblatex' package.

Cambridge University Press, Cambridge

titleGrassmannian geometry of scattering amplitudes

Springer Science
Business Media LLC

titleThe Amplituhedron
journaltitleJournal of High Energy Physics

titleZeroes of polynomials over finite fields
journaltitleAmer. J. Math.

titleMatroids, motives, and a conjecture of Kontsevich
journaltitleDuke Math. J.

pagesPaper No. 1.13, 20
titleQuotients of uniform positroids
journaltitleElectron. J. Combin.

titleOn motives associated to graph polynomials
journaltitleComm. Math. Phys.

titleKnots and numbers in $\phi^4$ theory to $7$ loops and
journaltitleInternat. J. Modern Phys. C

titleAssociation of multiple zeta values with positive knots via
Feynman diagrams up to $9$ loops
journaltitlePhys. Lett. B

pages181602, 4
titleDirect proof of the tree-level scattering amplitude recursion
relation in Yang-Mills theory
journaltitlePhys. Rev. Lett.

titleThe massless higher-loop two-point function
journaltitleComm. Math. Phys.

titleOn the periods of some Feynman integrals

Cambridge Univ. Press, Cambridge

booktitleGeometric and topological methods for quantum field
titleIterated integrals in quantum field theory

titleA K3 in $\phi^4$
journaltitleDuke Math. J.

titleModular forms in quantum field theory
journaltitleCommun. Number Theory Phys.

titleSingle-valued multiple polylogarithms and a proof of the
zig-zag conjecture
journaltitleJ. Number Theory

titleProperties of $c_2$ invariants of Feynman graphs
journaltitleAdv. Theor. Math. Phys.

titleSpanning forest polynomials and the transcendental weight of
Feynman graphs
journaltitleComm. Math. Phys.

journaltitleAnn. Inst. Henri Poincaré D

titleA combinatorial proof of the all minors matrix tree theorem
journaltitleSIAM J. Algebraic Discrete Methods

titleTableaux combinatorics for the asymmetric exclusion process
journaltitleAdv. in Appl. Math.

title$c_2$ invariants of recursive families of graphs
journaltitleAnn. Inst. Henri Poincaré D

titleOn one example and one counterexample in counting rational
points on graph hypersurfaces
journaltitleLett. Math. Phys.

titleThe $c_2$ invariant is invariant
journaltitleAdv. Theor. Math. Phys.

noteIn preperation
titleA result on the $c_2$ invariant for powers of primes

Cambridge University Press, Cambridge

isbn0-521-56144-2; 0-521-56724-6
noteWith applications to representation theory and geometry
seriesLondon Mathematical Society Student Texts
titleYoung tableaux

titleCombinatorial geometries, convex polyhedra, and Schubert
journaltitleAdv. in Math.

noteAccepted, to appear in Annales de l’Institut Henri
Poincaré D
titleFurther investigations into the graph theory of
$\phi^4$-periods and the $c_2$ invariant

McGraw-Hill International Book Co., New York

seriesInternational Series in Pure and Applied Physics
titleQuantum field theory

titleÜber die Auflösung der Gleichungen, auf welche man
bei der Untersuchung der linearen Vertheilung galvanischer Ströme
geführt wird
journaltitleAnnalen der Physik

titleSchubert calculus
journaltitleAmer. Math. Monthly

Springer, Heidelberg

booktitleAlgebras, quivers and representations
seriesAbel Symp.
titleCombinatorics of KP solitons from the real Grassmannian

journaltitleAnn. Inst. Henri Poincaré D

Springer, Berlin

booktitleMathematics unlimited—2001 and beyond

titleSome results on double triangle descendants of $K_5$
journaltitleAnn. Inst. Henri Poincaré D

titleThe positive tropical Grassmannian, the hypersimplex, and the
m=2 amplituhedron

Springer, Berlin

booktitleTopology and geometry—Rohlin Seminar
seriesLecture Notes in Math.
titleThe universality theorems on the classification problem of
configuration varieties and convex polytopes varieties

Oxford University Press, Oxford

seriesOxford Graduate Texts in Mathematics
titleMatroid theory

titleHepp's bound for Feynman graphs and matroids

titleTotal positivity, Grassmannians, and networks

World Sci. Publ., Hackensack, NJ

booktitleProceedings of the International Congress of
Mathematicians—Rio de Janeiro 2018. Vol. IV. Invited lectures
titlePositive Grassmannian and polyhedral subdivisions

titleThe m=2 amplituhedron and the hypersimplex: signs, clusters,
triangulations, Eulerian numbers

titleMatching polytopes, toric geometry, and the totally
non-negative Grassmannian
journaltitleJ. Algebraic Combin.

family=The Sage Developers,

titleSageMath, the Sage Mathematics Software System
(Version 8.2)

titleQuantum periods: a census of $\phi^4$-transcendentals
journaltitleCommun. Number Theory Phys.

pagesPaper 102, 23
titleQuantum field theory over $\mathbb{F}_q$
journaltitleElectron. J. Combin.

noteWith an appendix by F. Knop
titleGeometries in perturbative quantum field theory
journaltitleCommun. Number Theory Phys.

pagesPaper No. 100, 26
title$c_2$ invariants of hourglass chains via quadratic
denominator reduction
journaltitleSIGMA Symmetry Integrability Geom. Methods Appl.

titleBruhat interval polytopes
journaltitleAdv. Math.

titleA few $c_2$ invariants of circulant graphs
journaltitleCommun. Number Theory Phys.

Springer, Cham

isbn978-3-319-47550-9; 978-3-319-47551-6
seriesSpringerBriefs in Mathematical Physics
titleA combinatorial perspective on quantum field theory

howpublishedPrivate communications

titleA special case of completion invariance for the $c_2$
invariant of a graph
journaltitleCanad. J. Math.

EMS Publ. House, Berlin

booktitleAlgebraic combinatorics, resurgence, moulds and
applications (CARMA). Vol. 2
seriesIRMA Lect. Math. Theor. Phys.
titleA study on prefixes of $c_2$ invariants
year[2020] 2020
[segment=2,heading=subbibliography,title=Part I. The c2-invariant]

Missing 'biblatex' package
The bibliography requires the 'biblatex' package.

Cambridge University Press, Cambridge

titleGrassmannian geometry of scattering amplitudes

Springer Science
Business Media LLC

titleThe Amplituhedron
journaltitleJournal of High Energy Physics

titleZeroes of polynomials over finite fields
journaltitleAmer. J. Math.

titleMatroids, motives, and a conjecture of Kontsevich
journaltitleDuke Math. J.

pagesPaper No. 1.13, 20
titleQuotients of uniform positroids
journaltitleElectron. J. Combin.

titleOn motives associated to graph polynomials
journaltitleComm. Math. Phys.

titleKnots and numbers in $\phi^4$ theory to $7$ loops and
journaltitleInternat. J. Modern Phys. C

titleAssociation of multiple zeta values with positive knots via
Feynman diagrams up to $9$ loops
journaltitlePhys. Lett. B

pages181602, 4
titleDirect proof of the tree-level scattering amplitude recursion
relation in Yang-Mills theory
journaltitlePhys. Rev. Lett.

titleThe massless higher-loop two-point function
journaltitleComm. Math. Phys.

titleOn the periods of some Feynman integrals

Cambridge Univ. Press, Cambridge

booktitleGeometric and topological methods for quantum field
titleIterated integrals in quantum field theory

titleA K3 in $\phi^4$
journaltitleDuke Math. J.

titleModular forms in quantum field theory
journaltitleCommun. Number Theory Phys.

titleSingle-valued multiple polylogarithms and a proof of the
zig-zag conjecture
journaltitleJ. Number Theory

titleProperties of $c_2$ invariants of Feynman graphs
journaltitleAdv. Theor. Math. Phys.

titleSpanning forest polynomials and the transcendental weight of
Feynman graphs
journaltitleComm. Math. Phys.

journaltitleAnn. Inst. Henri Poincaré D

titleA combinatorial proof of the all minors matrix tree theorem
journaltitleSIAM J. Algebraic Discrete Methods

titleTableaux combinatorics for the asymmetric exclusion process
journaltitleAdv. in Appl. Math.

title$c_2$ invariants of recursive families of graphs
journaltitleAnn. Inst. Henri Poincaré D

titleOn one example and one counterexample in counting rational
points on graph hypersurfaces
journaltitleLett. Math. Phys.

titleThe $c_2$ invariant is invariant
journaltitleAdv. Theor. Math. Phys.

noteIn preperation
titleA result on the $c_2$ invariant for powers of primes

Cambridge University Press, Cambridge

isbn0-521-56144-2; 0-521-56724-6
noteWith applications to representation theory and geometry
seriesLondon Mathematical Society Student Texts
titleYoung tableaux

titleCombinatorial geometries, convex polyhedra, and Schubert
journaltitleAdv. in Math.

noteAccepted, to appear in Annales de l’Institut Henri
Poincaré D
titleFurther investigations into the graph theory of
$\phi^4$-periods and the $c_2$ invariant

McGraw-Hill International Book Co., New York

seriesInternational Series in Pure and Applied Physics
titleQuantum field theory

titleÜber die Auflösung der Gleichungen, auf welche man
bei der Untersuchung der linearen Vertheilung galvanischer Ströme
geführt wird
journaltitleAnnalen der Physik

titleSchubert calculus
journaltitleAmer. Math. Monthly

Springer, Heidelberg

booktitleAlgebras, quivers and representations
seriesAbel Symp.
titleCombinatorics of KP solitons from the real Grassmannian

journaltitleAnn. Inst. Henri Poincaré D

Springer, Berlin

booktitleMathematics unlimited—2001 and beyond

titleSome results on double triangle descendants of $K_5$
journaltitleAnn. Inst. Henri Poincaré D

titleThe positive tropical Grassmannian, the hypersimplex, and the
m=2 amplituhedron

Springer, Berlin

booktitleTopology and geometry—Rohlin Seminar
seriesLecture Notes in Math.
titleThe universality theorems on the classification problem of
configuration varieties and convex polytopes varieties

Oxford University Press, Oxford

seriesOxford Graduate Texts in Mathematics
titleMatroid theory

titleHepp's bound for Feynman graphs and matroids

titleTotal positivity, Grassmannians, and networks

World Sci. Publ., Hackensack, NJ

booktitleProceedings of the International Congress of
Mathematicians—Rio de Janeiro 2018. Vol. IV. Invited lectures
titlePositive Grassmannian and polyhedral subdivisions

titleThe m=2 amplituhedron and the hypersimplex: signs, clusters,
triangulations, Eulerian numbers

titleMatching polytopes, toric geometry, and the totally
non-negative Grassmannian
journaltitleJ. Algebraic Combin.

family=The Sage Developers,

titleSageMath, the Sage Mathematics Software System
(Version 8.2)

titleQuantum periods: a census of $\phi^4$-transcendentals
journaltitleCommun. Number Theory Phys.

pagesPaper 102, 23
titleQuantum field theory over $\mathbb{F}_q$
journaltitleElectron. J. Combin.

noteWith an appendix by F. Knop
titleGeometries in perturbative quantum field theory
journaltitleCommun. Number Theory Phys.

pagesPaper No. 100, 26
title$c_2$ invariants of hourglass chains via quadratic
denominator reduction
journaltitleSIGMA Symmetry Integrability Geom. Methods Appl.

titleBruhat interval polytopes
journaltitleAdv. Math.

titleA few $c_2$ invariants of circulant graphs
journaltitleCommun. Number Theory Phys.

Springer, Cham

isbn978-3-319-47550-9; 978-3-319-47551-6
seriesSpringerBriefs in Mathematical Physics
titleA combinatorial perspective on quantum field theory

howpublishedPrivate communications

titleA special case of completion invariance for the $c_2$
invariant of a graph
journaltitleCanad. J. Math.

EMS Publ. House, Berlin

booktitleAlgebraic combinatorics, resurgence, moulds and
applications (CARMA). Vol. 2
seriesIRMA Lect. Math. Theor. Phys.
titleA study on prefixes of $c_2$ invariants
year[2020] 2020
[segment=3,heading=subbibliography,title=Part II. Le diagrams]

Missing 'biblatex' package
The bibliography requires the 'biblatex' package.

Cambridge University Press, Cambridge

titleGrassmannian geometry of scattering amplitudes

Springer Science
Business Media LLC

titleThe Amplituhedron
journaltitleJournal of High Energy Physics

titleZeroes of polynomials over finite fields
journaltitleAmer. J. Math.

titleMatroids, motives, and a conjecture of Kontsevich
journaltitleDuke Math. J.

pagesPaper No. 1.13, 20
titleQuotients of uniform positroids
journaltitleElectron. J. Combin.

titleOn motives associated to graph polynomials
journaltitleComm. Math. Phys.

titleKnots and numbers in $\phi^4$ theory to $7$ loops and
journaltitleInternat. J. Modern Phys. C

titleAssociation of multiple zeta values with positive knots via
Feynman diagrams up to $9$ loops
journaltitlePhys. Lett. B

pages181602, 4
titleDirect proof of the tree-level scattering amplitude recursion
relation in Yang-Mills theory
journaltitlePhys. Rev. Lett.

titleThe massless higher-loop two-point function
journaltitleComm. Math. Phys.

titleOn the periods of some Feynman integrals

Cambridge Univ. Press, Cambridge

booktitleGeometric and topological methods for quantum field
titleIterated integrals in quantum field theory

titleA K3 in $\phi^4$
journaltitleDuke Math. J.

titleModular forms in quantum field theory
journaltitleCommun. Number Theory Phys.

titleSingle-valued multiple polylogarithms and a proof of the
zig-zag conjecture
journaltitleJ. Number Theory

titleProperties of $c_2$ invariants of Feynman graphs
journaltitleAdv. Theor. Math. Phys.

titleSpanning forest polynomials and the transcendental weight of
Feynman graphs
journaltitleComm. Math. Phys.

journaltitleAnn. Inst. Henri Poincaré D

titleA combinatorial proof of the all minors matrix tree theorem
journaltitleSIAM J. Algebraic Discrete Methods

titleTableaux combinatorics for the asymmetric exclusion process
journaltitleAdv. in Appl. Math.

title$c_2$ invariants of recursive families of graphs
journaltitleAnn. Inst. Henri Poincaré D

titleOn one example and one counterexample in counting rational
points on graph hypersurfaces
journaltitleLett. Math. Phys.

titleThe $c_2$ invariant is invariant
journaltitleAdv. Theor. Math. Phys.

noteIn preperation
titleA result on the $c_2$ invariant for powers of primes

Cambridge University Press, Cambridge

isbn0-521-56144-2; 0-521-56724-6
noteWith applications to representation theory and geometry
seriesLondon Mathematical Society Student Texts
titleYoung tableaux

titleCombinatorial geometries, convex polyhedra, and Schubert
journaltitleAdv. in Math.

noteAccepted, to appear in Annales de l’Institut Henri
Poincaré D
titleFurther investigations into the graph theory of
$\phi^4$-periods and the $c_2$ invariant

McGraw-Hill International Book Co., New York

seriesInternational Series in Pure and Applied Physics
titleQuantum field theory

titleÜber die Auflösung der Gleichungen, auf welche man
bei der Untersuchung der linearen Vertheilung galvanischer Ströme
geführt wird
journaltitleAnnalen der Physik

titleSchubert calculus
journaltitleAmer. Math. Monthly

Springer, Heidelberg

booktitleAlgebras, quivers and representations
seriesAbel Symp.
titleCombinatorics of KP solitons from the real Grassmannian

journaltitleAnn. Inst. Henri Poincaré D

Springer, Berlin

booktitleMathematics unlimited—2001 and beyond

titleSome results on double triangle descendants of $K_5$
journaltitleAnn. Inst. Henri Poincaré D

titleThe positive tropical Grassmannian, the hypersimplex, and the
m=2 amplituhedron

Springer, Berlin

booktitleTopology and geometry—Rohlin Seminar
seriesLecture Notes in Math.
titleThe universality theorems on the classification problem of
configuration varieties and convex polytopes varieties

Oxford University Press, Oxford

seriesOxford Graduate Texts in Mathematics
titleMatroid theory

titleHepp's bound for Feynman graphs and matroids

titleTotal positivity, Grassmannians, and networks

World Sci. Publ., Hackensack, NJ

booktitleProceedings of the International Congress of
Mathematicians—Rio de Janeiro 2018. Vol. IV. Invited lectures
titlePositive Grassmannian and polyhedral subdivisions

titleThe m=2 amplituhedron and the hypersimplex: signs, clusters,
triangulations, Eulerian numbers

titleMatching polytopes, toric geometry, and the totally
non-negative Grassmannian
journaltitleJ. Algebraic Combin.

family=The Sage Developers,

titleSageMath, the Sage Mathematics Software System
(Version 8.2)

titleQuantum periods: a census of $\phi^4$-transcendentals
journaltitleCommun. Number Theory Phys.

pagesPaper 102, 23
titleQuantum field theory over $\mathbb{F}_q$
journaltitleElectron. J. Combin.

noteWith an appendix by F. Knop
titleGeometries in perturbative quantum field theory
journaltitleCommun. Number Theory Phys.

pagesPaper No. 100, 26
title$c_2$ invariants of hourglass chains via quadratic
denominator reduction
journaltitleSIGMA Symmetry Integrability Geom. Methods Appl.

titleBruhat interval polytopes
journaltitleAdv. Math.

titleA few $c_2$ invariants of circulant graphs
journaltitleCommun. Number Theory Phys.

Springer, Cham

isbn978-3-319-47550-9; 978-3-319-47551-6
seriesSpringerBriefs in Mathematical Physics
titleA combinatorial perspective on quantum field theory

howpublishedPrivate communications

titleA special case of completion invariance for the $c_2$
invariant of a graph
journaltitleCanad. J. Math.

EMS Publ. House, Berlin

booktitleAlgebraic combinatorics, resurgence, moulds and
applications (CARMA). Vol. 2
seriesIRMA Lect. Math. Theor. Phys.
titleA study on prefixes of $c_2$ invariants
year[2020] 2020
[segment=4,heading=subbibliography,title=Part III. Final thoughts]