pith. sign in

arxiv: 2606.11137 · v2 · pith:Z6PLTS7Onew · submitted 2026-06-09 · 🧮 math.CO · math.AC

Enumeration of certain subsets of uprooted trees and spherical parking functions

Nayana Shibu Deepthi , Chanchal Kumar , Gargi Lather This is my paper

Pith reviewed 2026-06-27 12:27 UTC · model grok-4.3

classification 🧮 math.CO math.AC
keywords uprooted treesspherical parking functionsG-parking functionsmatrix tree theoremskeleton idealsenumerationlabeled treescomplete graphs
0
0 comments X

The pith

For graphs G_ℓ formed by deleting ℓ edges from the complete graph K_{n+1}, the number of uprooted trees on [n] with vertex 1 nonadjacent to F_ℓ equals (n-1)^{n-ℓ-2}(n-2)^ℓ(n-ℓ-1), and the number of spherical G_ℓ-parking functions equals (n-

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes explicit formulas for two related counting problems on a family of graphs G_ℓ obtained from K_{n+1} by removing the ℓ edges incident to vertex 1 and the set F_ℓ. It first counts the uprooted trees in which vertex 1 avoids all neighbors in F_ℓ, deriving the product formula both by direct combinatorial bijections and by an application of the matrix-tree theorem. These trees are shown to correspond to the spanning trees of G_ℓ minus an extra vertex. The same count is then used to determine the number of spherical G_ℓ-parking functions, defined as a distinguished subset of standard monomials arising from the skeleton ideals of the associated G-parking-function ideal. A reader interested in algebraic combinatorics would care because the formulas supply closed-form enumerations for parking-function variants on graphs that are only a few edges away from the complete graph, where such counts were previously known only in isolated cases.

Core claim

The uprooted spanning trees of G_ℓ minus vertex 0 stand in bijection with the set U_n^{1 not sim F_ℓ} of labeled uprooted trees on [n] in which vertex 1 shares no edge with any vertex of F_ℓ, and the cardinality of this set is (n-1)^{n-ℓ-2}(n-2)^ℓ(n-ℓ-1). As a direct consequence, the spherical G_ℓ-parking functions, which arise exactly as the distinguished standard monomials of the skeleton ideals of the G-parking-function ideal, are counted by (n-1)^{n-3}(n-ℓ-1)^2.

What carries the argument

The explicit bijection between the uprooted spanning trees of the punctured graph G_ℓ minus vertex 0 and the restricted set U_n^{1 not sim F_ℓ}, together with the identification of spherical parking functions as the distinguished standard monomials extracted from the skeleton ideals of the G-parking-function ideal.

If this is right

  • The matrix-tree theorem applied to the Laplacian of G_ℓ recovers the same product formula, yielding combinatorial identities as corollaries.
  • When ℓ=0 the formula reduces to the known count of all uprooted trees on n vertices.
  • The spherical-parking-function count is independent of the precise choice of which ℓ edges are deleted, depending only on the size ℓ.
  • The same skeleton-ideal construction produces spherical parking functions whose enumeration is given by a simple closed product for every member of the family G_ℓ.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same deletion construction and matrix-tree counting technique could be applied to graphs obtained by removing edges incident to more than one vertex.
  • The product formulas may admit natural q-analogues or generating-function lifts that encode additional statistics on the trees or parking functions.
  • Because the counts are explicit polynomials in n and ℓ, they supply exact asymptotics for large n with ℓ fixed or growing slowly with n.

Load-bearing premise

The uprooted spanning trees of G_ℓ minus vertex 0 are in exact bijection with the restricted uprooted trees on [n] avoiding edges from vertex 1 to F_ℓ, and spherical parking functions arise exactly as the distinguished subset of standard monomials from the skeleton ideals.

What would settle it

Direct enumeration of the restricted uprooted trees for n=4 and ℓ=1 should yield exactly (4-1)^{4-1-2}(4-2)^1(4-1-1)=3^1 * 2 * 2=12; any mismatch with the hand count of such trees on four labeled vertices would falsify the claimed formula.

Figures

Figures reproduced from arXiv: 2606.11137 by Chanchal Kumar, Gargi Lather, Nayana Shibu Deepthi.

Figure 1
Figure 1. Figure 1: General structure of a tree Te′ ∈ Be′ with sonTe′(r ′ ) = {a1, a2, . . . , as}. Each rectangle Fai represents the rooted forest of descendants of ai . As n−ℓ−2 s  s = (n − ℓ − 2) n−ℓ−3 s−1  , we have (2.5) |Be′ | = (n − ℓ − 1) (n − ℓ − 2) nX−ℓ−2 s=1  n − ℓ − 3 s − 1  (n − 2)n−s−3 . a1 a2 . . . as Fa1 Fa2 Fas [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The forest Te′ − {r ′}, where Te′ ∈ Be′ and sonTe′(r ′ ) = {a1, a2, . . . , as}. Thus, by combining (2.4) and (2.5), we have |B ′ | = (n − ℓ − 1)2 (n − ℓ − 2) nX−ℓ−2 s=1  n − ℓ − 3 s − 1  (n − 2)n−s−3 ! (By setting s − 1 = j) = (n − ℓ − 1)2 (n − ℓ − 2) (n − 2)ℓ−1   nX−ℓ−3 j=0  n − ℓ − 3 j  (n − 2)n−ℓ−3−j   = (n − ℓ − 1)2 (n − ℓ − 2) (n − 2)ℓ−1 (n − 1)n−ℓ−3 . This concludes the proof. □ Lemma 2.9. F… view at source ↗
Figure 3
Figure 3. Figure 3: ). If T ′′ 1 ∈ Ψ(U 1̸∼Fℓ n ), then we define τ (T ′′) = T ′′ 1 . Otherwise, replace T ′′ by T ′′ 1 (keeping the same root r ′′), and proceed to case (2). r ′′ Fr ′′ v Fv ea Fea ρ 1 Fρ −→ T ′′ T ′′ 1 r ′′ Fr ′′ v Fv ρ Fea ea 1 Fρ [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Construction of T ′′ 2 from the tree T ′′ . where cj ∈ {2, 3, . . . , n − ℓ} \ {r ′′} for all 1 ≤ j ≤ t, and w ∈ Fℓ \ {v} satisfies w < v. Without loss of generality, choose such a path for which w is minimal. We now construct a tree T ′′ 3 from T ′′ as follows. Remove the vertices w and v together with their descendant forests Fw and Fv, respectively. Then reattach w (together with Fw) as a son of the roo… view at source ↗
Figure 5
Figure 5. Figure 5: Construction of T ′′ 3 from the tree T ′′ . We now show that the above procedure is reversible. Let T ′ ∈ B′′ with root r ′ . We claim that there exists a tree T ′′ ∈ ` v∈Fℓ Tv such that τ (T ′′) = T ′ . Suppose that r ′ is adjacent to a vertex v ∈ Fℓ . Then, by Proposition 2.4, the root r ′ is not adjacent to any other vertex of Fℓ . Starting from T ′ , we reconstruct a tree T ′′ with the required propert… view at source ↗
Figure 6
Figure 6. Figure 6: Construction of T ′′ from the tree Te, where root (T ′′) ∈ sonTe (v)\{aj}. Therefore, by Theorem 2.2, the number of trees T with sonT (v) = {a1, a2, . . . , as}, as de￾scribed above, is s(n−2)n−3−s . The number of ways to choose a subset sonT (v) ⊆ {2, 3, . . . , n−ℓ} with s elements is n−ℓ−1 s  . For a fixed tree T, we construct a tree Te by attaching the vertex 1 as a leaf. Since 1 may be attached to an… view at source ↗
Figure 7
Figure 7. Figure 7: Possible structures of the rooted subtree Fv(T). For the maximum path Pℓ max(T) : a0 → a1 → · · · → aα, let F(ai) (T) denote the rooted forest obtained from Fai (T) by removing the subtree rooted at ai+1. Equivalently, the roots of the connected components of F(ai) (T) are precisely the sons of ai other than ai+1. Let u and v be vertices of the rooted tree T such that u is a leaf of T and is not a descenda… view at source ↗
Figure 8
Figure 8. Figure 8: The operation T ′ = T [PITH_FULL_IMAGE:figures/full_fig_p022_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: A tree T ∈ Cp n,ℓ with maximum path Pℓ max(T) : r → a1 → a2 → · · · → aα and the associated forests. Proposition 4.5. Let 0 ≤ p ≤ ℓ − 2, and let T ∈ Un be an uprooted tree with root(T) = n − p. Suppose that T has the following properties. (1) The maximum path P ℓ max(T) of T in Fℓ is of the form n−p = a0 → a1 → · · · → aα−1 → aα, where α ≥ 1 and a1 < n − p, (2) vertex 1 is a leaf of T and is not adjacent t… view at source ↗
Figure 10
Figure 10. Figure 10: The structure of a tree Te ∈ Tn,ℓ. Proposition 5.1. The cardinality of Tn,ℓ is given by |Tn,ℓ| = (n − 1)n−ℓ−2 (n − 2)ℓ−2 (n − ℓ − 1) [PITH_FULL_IMAGE:figures/full_fig_p024_10.png] view at source ↗
Figure 10
Figure 10. Figure 10: Let S = {i1, i2, . . . , is} be the set of roots of the connected components of the forest F(r) (Te). Then ∅ ̸= S ⊆ {2, 3, . . . , n − ℓ}. Therefore, the set S with s elements can be chosen in [PITH_FULL_IMAGE:figures/full_fig_p025_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: The tree Te2 with path Q′ and the associated descendant forests. We now divide Case III into two subcases [PITH_FULL_IMAGE:figures/full_fig_p026_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Illustration of the tree Te3 . Next, remove the forests F(b ′ j ) (Te2 ) for 1 ≤ j ≤ t from Te3 . Then attach F(b ′ 1 ) (Te2 ) to n − ℓ + 1, and, for each 1 < j ≤ t, attach F(b ′ j ) (Te2 ) to b ′ j−1 , thereby obtaining Te4 (see [PITH_FULL_IMAGE:figures/full_fig_p027_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: The tree Te4 [PITH_FULL_IMAGE:figures/full_fig_p027_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: The structure of the tree Te5 . If sonTe(1) ∩ Fℓ = ∅, then Te5 ∈ Cℓ−2 n,ℓ , and we define ψ(Te) = Te5 . Otherwise, suppose that sonTe(1) ∩ Fℓ ̸= ∅. In this case, we consider Te6 = Te5 [PITH_FULL_IMAGE:figures/full_fig_p028_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Illustration of the rooted subtree F(b ′ p ) (Te2 ) = b ′ p ∨ F(b ′ p ) (Te2 ) [PITH_FULL_IMAGE:figures/full_fig_p028_15.png] view at source ↗
Figure 16
Figure 16. Figure 16: The structure of the tree Te7 . Observe that F1(Te7 ) = F1(Te2 ). If sonTe7 (1)∩Fℓ = ∅, then Te7 ∈ Cℓ−2 n,ℓ , and we define ψ(Te) = Te7 . Otherwise, if sonTe7 (1) ∩ Fℓ ̸= ∅, then we consider the tree Te8 = Te7 [PITH_FULL_IMAGE:figures/full_fig_p029_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Structure of the rooted subtree F(at) (T 3 ) in T 3 . Let Fat+1 (T 3 ) denote the rooted forest consisting of all descendants of at+1 in T 3 . Now consider the tree T 4 = T 3 [PITH_FULL_IMAGE:figures/full_fig_p030_17.png] view at source ↗
Figure 18
Figure 18. Figure 18: Illustration of the structure of T 4 . Next, we remove the forests F ′ (at) (T 3 ), Fa ′ 1 (T 3 ), and F(ai) (T 3 ) for 1 ≤ i < t from T 4 . Then attach F ′ (at) (T 3 ) and Fa ′ 1 (T 3 ) to at+1. Then, for each 1 ≤ i < t, attach F(ai) (T 3 ) to ai+1. The resulting tree T 5 is illustrated in [PITH_FULL_IMAGE:figures/full_fig_p030_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: The tree T 5 . Now, consider the rooted subtree Fa2 (T 5 ), consisting of the vertex a2 together with all its descendants in T 5 . By removing Fa2 (T 5 ) from T 5 and attaching it to the root r = n − ℓ + 2 in T 5 , we obtain T 6 , illustrated in [PITH_FULL_IMAGE:figures/full_fig_p031_19.png] view at source ↗
Figure 20
Figure 20. Figure 20: The tree T 6 . Finally, by interchanging the vertices a2 and a ′ q in T 6 , we obtain a tree T 7 ∈ Tn,ℓ. Reversing the above constructions, we see that ψ(T 7 ) = T. Subclass (2). Suppose that the son of at = a ′ 0 lying on the path Q′ belongs to Fℓ ; that is, a ′ 1 ∈ Fℓ . Proceeding as in Subclass (1), we first construct the tree T 4 . From T 4 , we then construct a tree T 8 of the form illustrated in [P… view at source ↗
Figure 21
Figure 21. Figure 21: The tree T 8 . (1) If α = 1, then T arises from Case I. (2) If α > 1 and 1 is a descendant of a1 = n − ℓ + 1, then T arises from Case II. (3) Suppose α > 1 and there exists 1 ≤ t < α such that 1 is a descendant of at but not of at+1, then T arises from Case III. Let Q′ : at = a ′ 0 → a ′ 1 → · · · → a ′ q → 1 be the unique path from at to 1. Then: • if a ′ 1 ∈/ Fℓ , then T arises from Subcase III(a), • if… view at source ↗
Figure 22
Figure 22. Figure 22: r = n − p n − ℓ + 1 F ′ (r) (Te4 ) Fn−ℓ+1(Te4 ) F1(Te) (a) The tree Te4 obtained from Te. c0 = r = n − p c1 c2 cγ F(1)(Te) F(c1)(Te) F(c2)(Te) F(cγ )(Te) F1(Te) (b) The rooted tree r ∨ F1(Te). r = n − p b1 b2 bβ F ′′ (r) (Te) F(b1)(Te) F(b2)(Te) F(bβ)(Te) F ′ (r) (Te4 ) (c) The rooted tree r ∨ F′ (r) (Te4 ) where 1 is a leaf which is not a descendant of bβ [PITH_FULL_IMAGE:figures/full_fig_p036_22.png] view at source ↗
read the original abstract

Spherical $G$-parking functions are a distinguished subset of standard monomials, arising from the skeleton ideals of the $G$-parking function ideal. Explicit enumeration formulas for spherical $G$-parking functions are known only for a few classes of graphs. In this paper, we consider a family of graphs $G_{\ell}$ ($1\leq \ell \leq n-2$), obtained from the complete graph $K_{n+1}$ by deleting the $\ell$ edges joining vertex $1$ to the vertices in $F_{\ell}= \{n-\ell+1, \ldots, n\}$. The uprooted spanning trees of $G_{\ell}-\{0\}$ correspond to the set $\mathcal{U}_n^{1\not\sim F_{\ell}}$ of uprooted trees with vertex set $[n]$ in which vertex $1$ is not adjacent to any vertex in $F_{\ell}$, and we establish that $|\mathcal{U}_n^{1\not\sim F_{\ell}}| = (n-1)^{n-\ell-2}(n-2)^{\ell}(n-\ell-1)$. We derive this formula combinatorially and independently recover it as an application of the matrix tree theorem, obtaining some combinatorial identities as consequences. Finally, we determine the number of spherical $G_{\ell}$-parking functions as $|\mathrm{sPF}(G_{\ell})| = (n-1)^{n-3}(n-\ell-1)^2$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 0 minor

Summary. The paper claims to enumerate the set Τ_n^{1≁F_ℓ} of uprooted trees on [n] in which vertex 1 is not adjacent to any vertex in F_ℓ, establishing the closed form (n-1)^{n-ℓ-2}(n-2)^ℓ(n-ℓ-1) via a direct combinatorial argument; it independently recovers the same count by applying the matrix-tree theorem to the Laplacian of G_ℓ−{0}. It then determines the number of spherical G_ℓ-parking functions as (n-1)^{n-3}(n-ℓ-1)^2.

Significance. If the formulas held, the work would supply explicit enumerations for spherical parking functions on the family of graphs obtained from K_{n+1} by deleting ℓ edges incident to a fixed vertex, together with combinatorial identities arising from the matrix-tree application. The dual combinatorial and algebraic derivations would be a methodological strength.

major comments (1)
  1. [Abstract] Abstract: the claimed formula |U_n^{1≁F_ℓ}| = (n-1)^{n-ℓ-2}(n-2)^ℓ(n-ℓ-1) is inconsistent with known counts. When ℓ=0 the expression reduces to (n-1)^{n-1}, whereas G_0−{0} is K_n and the number of its spanning trees is n^{n-2} by Cayley's formula. For the concrete case n=4, ℓ=1 the formula yields 12, but K_4 minus one edge has exactly 8 spanning trees (equivalently 2n^{n-3}=8). Because the paper asserts both a combinatorial derivation and an independent matrix-tree recovery of this same expression, the mismatch shows that either the bijection with spanning trees of G_ℓ−{0} or the closed-form derivation itself is incorrect; this error is load-bearing for both the tree enumeration and the subsequent spherical-parking-function count.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their detailed review and for identifying the inconsistency between the claimed formula and standard results such as Cayley's formula. We agree that the stated enumeration is incorrect and that this affects the core claims of the paper. We will revise the manuscript to correct the formula and the subsequent parking-function count.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the claimed formula |U_n^{1≁F_ℓ}| = (n-1)^{n-ℓ-2}(n-2)^ℓ(n-ℓ-1) is inconsistent with known counts. When ℓ=0 the expression reduces to (n-1)^{n-1}, whereas G_0−{0} is K_n and the number of its spanning trees is n^{n-2} by Cayley's formula. For the concrete case n=4, ℓ=1 the formula yields 12, but K_4 minus one edge has exactly 8 spanning trees (equivalently 2n^{n-3}=8). Because the paper asserts both a combinatorial derivation and an independent matrix-tree recovery of this same expression, the mismatch shows that either the bijection with spanning trees of G_ℓ−{0} or the closed-form derivation itself is incorrect; this error is load-bearing for both the tree enumeration and the subsequent spherical-parking-function count.

    Authors: We concur with the referee's observation. The formula does not reduce to n^{n-2} when ℓ=0, nor does it match the known count of 8 spanning trees for the n=4, ℓ=1 case. This demonstrates that at least one of the two derivations (combinatorial or matrix-tree) contains an error. We will revise the manuscript by deriving the correct closed form for |U_n^{1≁F_ℓ}| (and the induced count for |sPF(G_ℓ)|) via a verified application of the matrix-tree theorem to the Laplacian of the appropriate graph, together with any necessary corrections to the combinatorial argument. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external matrix-tree theorem and claimed combinatorial bijection.

full rationale

The paper claims a bijection between uprooted spanning trees of G_ℓ−{0} and U_n^{1≁F_ℓ}, then states the enumeration formula is derived combinatorially and independently recovered via the matrix tree theorem on the Laplacian of the graph (an external, non-self-referential result). The spherical parking-function count is obtained from this tree enumeration using the algebraic definition of spherical G-parking functions as a subset of standard monomials. No step reduces a claimed prediction to a fitted input, self-citation, or definitional equivalence within the paper; the central claims rest on independent external tools and the stated bijection rather than internal redefinition.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The claims rest on the standard matrix tree theorem (an external result) and on the stated correspondence between the tree set and the uprooted trees of the modified graph; no free parameters or new entities are introduced in the abstract.

axioms (1)
  • standard math Matrix tree theorem counts the spanning trees of a graph via any cofactor of its Laplacian matrix.
    Invoked to independently recover the enumeration formula for the uprooted trees.

pith-pipeline@v0.9.1-grok · 5802 in / 1489 out tokens · 23606 ms · 2026-06-27T12:27:59.646519+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

21 extracted references · 1 linked inside Pith

  1. [1]

    Aigner,A course in enumeration, Graduate Texts in Mathematics, vol

    M. Aigner,A course in enumeration, Graduate Texts in Mathematics, vol. 238, Springer, 2007

    2007

  2. [2]

    Backman, C

    S. Backman, C. Charbonneau, N. A. Loehr, P. Mullins, M. O’Connor, and G. S. Warrington,Skeletal gener- alizations of chip-firing games, parking functions, and Dyck paths, Combin. Theory5(2025), no. 4, #6

    2025

  3. [3]

    R. B. Bapat,Graphs and matrices, Universitext, Springer-Verlag, London, 2010

    2010

  4. [4]

    Chauve, S

    C. Chauve, S. Dulucq, and O. Guibert,Enumeration of some labelled trees, In: D. Krob, A. A. Mikhalev, and A. V. Mikhalev (eds.), Formal Power Series and Algebraic Combinatorics, Springer, Berlin, 2000, pp. 146– 157

    2000

  5. [5]

    Chebikin and P

    D. Chebikin and P. Pylyavskyy,A family of bijections betweenG-parking functions and spanning trees, J. Combin. Theory Ser. A110(2005), no. 1, 31–41

    2005

  6. [6]

    N. S. Deepthi and C. Kumar,Combinatorial identities using the matrix tree theorem, Ars Comb.165(2025), 45–61

    2025

  7. [7]

    Dhar,Self-organized critical state of sandpile automaton models, Phys

    D. Dhar,Self-organized critical state of sandpile automaton models, Phys. Rev. Lett.64(1990), no. 14, 1613– 1616

    1990

  8. [8]

    Dochtermann,One-skeleta ofG-parking function ideals: resolutions and standard monomials, arXiv:1708.04712 (2017)

    A. Dochtermann,One-skeleta ofG-parking function ideals: resolutions and standard monomials, arXiv:1708.04712 (2017)

    Pith/arXiv arXiv 2017

  9. [9]

    Dochtermann and W

    A. Dochtermann and W. King,Trees, parking functions, and standard monomials of skeleton ideals, Australas. J. Combin.81(2021), no. 1, 126–151

    2021

  10. [10]

    Gaydarov and S

    P. Gaydarov and S. Hopkins,Parking functions and tree inversions revisited, Adv. Appl. Math.80(2016), 151–179

    2016

  11. [11]

    Guedes de Oliveira and M

    A. Guedes de Oliveira and M. Las Vergnas,Parking functions and labeled trees, S´ em. Lothar. Combin.65 (2011), B65e

    2011

  12. [12]

    Kirchhoff,Ueber die aufl¨ osung der gleichungen, auf welche man bei der untersuchung der linearen ver- theilung galvanischer str¨ ome gef¨ uhrt wird, Ann

    G. Kirchhoff,Ueber die aufl¨ osung der gleichungen, auf welche man bei der untersuchung der linearen ver- theilung galvanischer str¨ ome gef¨ uhrt wird, Ann. Phys.148(1847), 497–508. (English transl. IRE Trans. Circuit Theory CT-5 (1958), 4-7.)

    1958

  13. [13]

    A. G. Konheim and B. Weiss,An occupancy discipline and applications, SIAM J. Appl. Math.14(1966), no. 6, 1266–1274

    1966

  14. [14]

    Kreweras,Une famille de polynˆ omes ayant plusieurs propri´ et´ es ´ enumeratives, Period

    G. Kreweras,Une famille de polynˆ omes ayant plusieurs propri´ et´ es ´ enumeratives, Period. Math. Hung.11 (1980), 309–320

    1980

  15. [15]

    Kumar, G

    C. Kumar, G. Lather, and A. Roy,Standard monomials of 1-skeleton ideals of graphs and generalized signless Laplacians, Linear Algebra Appl.637(2022), 24–48. ENUMERATION OF UPROOTED TREES AND SPHERICAL PARKING FUNCTIONS 43

    2022

  16. [16]

    Kumar, G

    C. Kumar, G. Lather, and Sonica,Skeleton ideals of certain graphs, standard monomials and spherical parking functions, Electron. J. Comb.28(2021), no. 1, #P1.53

    2021

  17. [17]

    J. W. Moon,Counting labelled trees, Canadian Mathematical Monographs, vol. 1, Canadian Mathematical Congress, Montreal, 1970

    1970

  18. [18]

    Perkinson, Q

    D. Perkinson, Q. Yang, and K. Yu,G-parking functions and tree inversions, Combinatorica37(2017), no. 2, 269–282

    2017

  19. [19]

    Postnikov and B

    A. Postnikov and B. Shapiro,Trees, parking functions, syzygies, and deformations of monomial ideals, Trans. Amer. Math. Soc.356(2004), no. 8, 3109–3142

    2004

  20. [20]

    Riordan,Ballots and trees, J

    J. Riordan,Ballots and trees, J. Combin. Theory6(1969), no. 4, 408–411

    1969

  21. [21]

    R. P. Stanley,Hyperplane arrangements, interval orders, and trees, Proc. Natl. Acad. Sci. U.S.A.93(1996), no. 6, 2620–2625. Department of Mathematical Sciences, IISER Mohali, Knowledge City, Sector 81, SAS Nagar, Punjab, 140-306, India Email address:nayanashibu@iisermohali.ac.in Department of Mathematical Sciences, IISER Mohali, Knowledge City, Sector 81,...

    1996