Skeleton Ideals of Certain Graphs, Standard Monomials and Spherical Parking Functions

Let $G$ be an (oriented) graph on the vertex set $V = \{ 0, 1,\ldots,n\}$ with root $0$. Postnikov and Shapiro associated a monomial ideal $\mathcal{M}_G$ in the polynomial ring $ R = {\mathbb{K}}[x_1,\ldots,x_n]$ over a field $\mathbb{K}$. A subideal $\mathcal{M}_G^{(k)}$ of $\mathcal{M}_G$ generated by subsets of $\widetilde{V}=V\setminus \{0\}$ of size at most $k+1$ is called a $k$-skeleton ideal of the graph $G$. Many interesting homological and combinatorial properties of $1$-skeleton ideal $\mathcal{M}_G^{(1)}$ are obtained by Dochtermann for certain classes of simple graph $G$. A finite sequence $\mathcal{P}=(p_1,\ldots,p_n) \in \mathbb{N}^n$ is called a spherical $G$-parking function if the monomial $\mathbf{x}^{\mathcal{P}} = \prod_{i=1}^{n} x_i^{p_i} \in \mathcal{M}_G \setminus \mathcal{M}_G^{(n-2)}$. Let ${\rm sPF}(G)$ be the set of all spherical $G$-parking functions. In this paper, a combinatorial description for all multigraded Betti numbers of the $k$-skeleton ideal $\mathcal{M}_{K_{n+1}}^{(k)}$ of the complete graph $K_{n+1}$ on $V$ are given. Also, using DFS burning algorithms of Perkinson-Yang-Yu (for simple graph) and Gaydarov-Hopkins (for multigraph), we give a combinatorial interpretation of spherical $G$-parking functions for the graph $G = K_{n+1}- \{e\}$ obtained from the complete graph $K_{n+1}$ on deleting an edge $e$. In particular, we showed that $|{\rm sPF}(K_{n+1}- \{e_0\} )|= (n-1)^{n-1}$ for an edge $e_0$ through the root $0$, but $|{\rm sPF}(K_{n+1} - \{e_1\})| = (n-1)^{n-3}(n-2)^2$ for an edge $e_1$ not through the root.


Introduction
Let G be an oriented graph on the vertex set V = {0, 1, . . . , n} with a root 0. The graph G is completely determined by an (n + 1) × (n + 1) matrix A(G) = [a ij ] 0≤i,j≤n , called its adjacency matrix, where a ij is the number of oriented edges from i to j. A non-oriented graphG on V (rooted at 0) is identified with the unique (rooted) oriented graph G on V with symmetric adjacency matrix A(G) = A(G). Let R = K[x 1 , . . . , x n ] be the standard polynomial ring in n variables over a field K. The G-parking function ideal M G of G is a monomial ideal in R given by the generating set where d A (i) = j∈V \A a ij is the number of edges from i to a vertex outside the set A in G. A standard monomial basis {x b = n i=1 x b i i } of the the Artinian quotient R M G is determined by the set PF(G) = {b = (b 1 , . . . , b n ) ∈ N n : x b / ∈ M G } of G-parking functions. Further, dim K R M G is the number of (oriented) spanning trees of G, given by the determinant det(L G ) of the truncated Laplace matrix L G of G. Let SPT(G) be the set of (oriented) spanning trees of G. If G is nonoriented, then the edges of a spanning tree of G is given orientation so that all paths in the spanning tree are directed away from the root. In this paper, G is assumed to be a non-oriented graph. For G = K n+1 , the complete graph on V , the K n+1 -parking functions are the ordinary parking functions of length n. As |PF(G)| = |SPT(G)|, one would like to construct an explicit bijection φ : PF(G) −→ SPT(G). Using a depth-first search (DFS) version of burning algorithm, an algorithmic bijection φ : PF(G) −→ SPT(G) for simple graphs G preserving reverse (degree) sum of G-parking function P and the number of κ-inversions of the spanning tree φ(P) is constructed by Perkinson, Yang and Yu [12]. A similar bijection for multigraphs G is constructed by Gaydarov and Hopkins [5].
Postnikov and Shapiro [14] introduced the ideal M G and derived many of its combinatorial and homological properties. In particular, they showed that the minimal resolution of M G is the cellular resolution supported on the first barycentric subdivision Bd(∆ n−1 ) of an n − 1-simplex ∆ n−1 , provided G is saturated (i.e., a ij > 0 for i = j). The minimal resolution of M G for any graph G is described in [2,9,11].
Dochtermann [3,4] proposed to investigate subideals of the G-parking function ideal M G described by k-dimensional 'skeleta'. For an integer k (0 ≤ k ≤ n − 1), the k-skeleton ideal M G is generated by powers of variables x 1 , . . . , x n . Hence, its minimal free resolution and the number of standard monomials can be easily determined. For k = 1 and G = K n+1 , the minimal resolution of one-skeleton ideal M (1) K n+1 is a cocellular resolution supported on the labelled polyhedral complex induced by any generic arrangement of two tropical hyperplanes in R n and i th Betti number β i R M (1) K n+1 = n j=1 j j−1 i−1 for 1 ≤ i ≤ n − 1 (see [4]). Also, the number of standard monomials of R M (1) K n+1 is given by where Q K n+1 is the truncated signless Laplace matrix of K n+1 . A notion of spherical G-parking functions is introduced in [3] for the complete graph G = K n+1 . Let PF(G) = {P ∈ N n : x P / ∈ M G } be the set of G-parking functions. Consider the set sPF(G) = {P ∈ N n : x P ∈ M G \M finite sequence P = (p 1 , . . . , p n ) ∈ sPF(G) is called a spherical G-parking function. A G-parking or a spherical G-parking function P = (p 1 , . . . , p n ) ∈ N n can be equivalently thought of as a function P : The sum (or degree) of P is given by sum(P) = i∈ V P(i). We recall that a K n+1 -parking function P = (p 1 , . . . , p n ) ∈ N n is an ordinary parking function of length n, i.e., a non-decreasing rearrangement p i 1 ≤ p i 2 ≤ . . . ≤ p in of P = (p 1 , . . . , p n ) satisfies p i j < j, ∀j. It can be easily checked that P = (p 1 , . . . , p n ) ∈ N n is a spherical K n+1 -parking function if a non-decreasing rearrangement p i 1 ≤ p i 2 ≤ . . . ≤ p in of P = (p 1 , . . . , p n ) satisfies p i 1 = 1 and p i j < j for 2 ≤ j ≤ n. The notion of spherical K n+1 -parking function has appeared earlier in the literature (see [15]) as prime parking functions of length n. Prime parking functions were defined and enumerated by I. Gessel. The number of spherical K n+1parking functions is (n − 1) n−1 , which is same as the number of uprooted trees on [n]. A (labelled) rooted tree T on [n] is called uprooted if the root is bigger than all its children. Let U n be the set of uprooted trees on [n]. Dochtermann [3] conjectured existence of a bijection φ : sPF(K n+1 ) −→ U n such that sum(P) = n 2 − κ(K n , φ(P)) + 1, where κ(K n , φ(P)) is the κ-number of the uprooted tree φ(P) in the complete graph The k-skeleton ideal M (k) K n+1 of the complete graph K n+1 can be identified with an Alexander dual of some multipermutohedron ideal. Let u = (u 1 , u 2 , . . . , u n ) ∈ N n such that u 1 ≤ u 2 ≤ . . . ≤ u n . Set m = (m 1 , . . . , m s ) such that the smallest entry in u is repeated exactly m 1 times, second smallest entry in u is repeated exactly m 2 times, and so on. Then s j=1 m j = n and m j ≥ 1 for all j. In this case, we write u(m) for u. Let S n be the set of permutations of [n]. For a permutation σ of . The monomial ideal I(u(m)) = x σu(m) : σ ∈ S n of R is called a multipermutohedron ideal. If m = (1, . . . , 1) ∈ N n , then I(u(m)) is called a permutohedron ideal.
Let u(m) = (1, 2, . . . , k, k + 1, . . . , k + 1) ∈ N n , where m = (1, . . . , 1, n − k) ∈ N k+1 . For 1 ≤ k ≤ n − 1, the Alexander dual I(u(m)) [n] of multipermutohedron ideal I(u(m)) with respect to n = (n, . . . , n) ∈ N n equals the k-skeleton ideal M (k) K n+1 . Thus, the number of standard monomials of the Artinian quotient is given by the number of λ-parking functions for λ = (n, n − 1, . . . , n − k + 1, n − k, . . . , n − k) ∈ N n (see [13,14]). A description of multigraded Betti numbers of Alexander duals of multipermutohedron ideals and a simple proof of the Steck determinant formula for the number of λ-parking functions are given in [7]. We obtain a combinatorial expression for (Proposition 2.2). In particular, for n ≥ 3, we show that By a simple modification of DFS algorithm of Peterson, Yang and Yu [12], we construct an algorithmic bijection φ n : sPF(K n+1 ) −→ U n as conjectured by Dochtermann [3]. Further, we compare spherical parking functions of a graph G with that of G − {e}. If e is an edge of G, then G − {e} is the graph obtained on deleting the edge e from G. We show that |sPF(G)| = |sPF(G−{e 0 })| (Lemma 3.1), where e 0 is an edge from the root to another vertex. As an application of this result, we show that the number of spherical parking functions of a complete bipartite graph K m+1,n satisfies We obtain a formula for |sPF(K m+1,n )| (Proposition 3.4), which is symmetric in m and n. For the complete graph K n+1 and an edge e 1 not through the root, we show that |sPF(K n+1 − {e 1 })| = (n − 1) n−3 (n − 2) 2 (Theorem 3.10).
Some extensions of these results for the complete multigraph K a,b n+1 are also obtained.

k-skeleton ideals of complete graphs
Let 0 ≤ k ≤ n − 1. Consider the k-skeleton ideal M  K n+1 = x n 1 , . . . , x n n is a monomial ideal in R generated by n th power of variables. Thus, its minimal free resolution is given by the Koszul complex associated to the regular sequence The minimal free resolution of the K n+1 -parking function ideal M K n+1 is the cellular resolution supported on the first barycentric subdivision Bd(∆ n−1 ) of an n − 1-simplex ∆ n−1 and dim K K n+1 has a minimal cocellular resolution supported on the labelled polyhedral complex induced by any generic arrangement of two tropical hyperplanes in R n−1 (see Theorem 4.6 of [4]) and dim K Betti numbers of M Proof. Using Proposition 5.23 of [10], it follows from the Lemma 2.3 of [7].
The multigraded Betti numbers of Alexander duals of a multipermutohedron ideal are described in terms of dual m-isolated subsets (see Definition 3.1 and Theorem 3.2 of [7]). For the particular case of m = (1, . . . , 1, n − k) ∈ N k+1 , the notion of dual m-isolated subsets can be easily described. Let J = {j 1 , . . . , j t } ⊆ [n] be a non-empty subset with 0 = j 0 < j 1 < . . . < j t .
(1) J is a dual m-isolated subset of type-1 if J ⊆ [k + 1] and its dual weight dwt(J) = t − 1. Let I * ,1 m be the set of dual m-isolated subsets of type-1 and let I * ,1 m i = {J ∈ I * ,1 m : dwt(J) = i}.
Let e 1 , e 2 , . . . , e n be the standard basis vectors of R n . For 0 ≤ i < j ≤ n, we set ε(i, j) = j l=i+1 e l . For any Then the following statements hold.
(i) For J = {j 1 , . . . , j t } ∈ I * ,1 is a permutation of b(J) for some J ∈ I * m i − 1 and some π ∈ S n , then K n+1 . We hope that it could be helpful in constructing a concrete minimal resolution of M K n+1 is precisely the number of λ-parking functions for λ = (λ 1 , . . . , λ n ), where λ i = n − i + 1 for 1 ≤ i ≤ k and λ j = n − k for k + 1 ≤ j ≤ n (see [14]).
Ordinary parking functions of length n are precisely λ-parking functions of length n for λ = (n, n − 1, . . . , 2, 1) ∈ N n . The number of λ-parking functions is given by the following Steck . In other words, the (i, j) th entry of the n × n matrix Λ(λ 1 , . . . , λ n ) is We proceed to evaluate Steck determinant and compute the number of standard monomials of M (k) K n+1 as indicated in [8]. For more on parking functions, we refer to [13,14,16]. More generally, for a, b ≥ 1, we consider the complete multigraph K a,b n+1 on the vertex set V with adjacency matrix A(K a,b n+1 ) = [a ij ] 0≤i,j≤n given by a 0,i = a i,0 = a and a i, n+1 has exactly a number of edges between the root 0 and any other vertex i, while it has exactly b number of edges between distinct non-rooted vertices i and j. Clearly, Also, for 1 ≤ k ≤ n − 2, consider another polynomial g n;k (x) in x given by Proposition 2.5. The polynomials f n (x) and g n;k (x) are given as follows. ( Further, using properties of determinants, we observe that the by parts, we get (1).
Again using properties of determinants, we see that the (n − k − 1) th derivative g are given as follows. ( In particular, for k = 1 and k = n − 2, In view of Proposition 2.5, we get (1) and (2).
Also, for k = n − 2, we have g n;n−2 ( On integrating it by parts, where C is a constant of integration. Since g n;n−2 (0) = 0, we get C = (n−1) n−1 b n n!
(1) It can be easily checked that the determinant det(Q K a,b n+1 ) of the truncated ).
This extends Corollary 3.4 of [4] to the complete multigraph K a,b n+1 .
Thus on integrating g n;n−2 (x) in two ways, we get a polynomial identity On substituting x = a + b, we get an identity n+1 -parking function is given by |sPF(K a,b n+1 )| = (n − 1) n−1 b n . Note that this number is independent of a.
DFS burning Algorithm : We shall briefly describe Depth-First-Search (DFS) burning algorithms of Perkinsons-Yang-Yu [12] and Gaydarov-Hopkins [5]. Firstly, we set up graph theoretic notations and invariants needed for the DFS algorithm. Let G be a connected graph on the vertex set Since G has no loops, a ii = 0. Let E(i, j) = E(j, i) be the set of edges between i and j for distinct i, j ∈ V . If E(i, j) = ∅, then i and j are called adjacent vertices and we write i ∼ j. On the other hand, if i and j are non-adjacent, we write i j. We have |E(i, j)| = a ij . The graph G is called a simple graph if |E(i, j)| = a ij ≤ 1 for i, j ∈ V . Otherwise, G is called a multigraph. The set E(G) = i,j∈V E(i, j) is the set of edges of the graph G. If v ∈ V , then G − {v} denotes the graph on the vertex set V \ {v} obtained from G on deleting the vertex v and all the edges through v. If e ∈ E(G) is an edge of G, then G − {e} denotes the graph on the vertex set V obtained from G on deleting the edge e. If E(i, j) = ∅, then G − E(i, j) denotes the graph on vertex set V obtained from G on deleting all the edges between i and j. Fix a root r ∈ V of G (usually, we take r = 0). Set V = V \ {r}. Let SPT(G) be the set of spanning trees of G rooted at r. We orient spanning tree T ∈ SPT(G) so that all paths in T are directed away from the root r. For every j ∈ V , there is a unique oriented path in T from the root r to j. An i ∈ V lying on this unique path in T is called an ancestor of j in T . Equivalently, we say that j is a descendent of i in T . If in addition, i and j are adjacent in T , then we say that i is a parent of its child j. Every child j has a unique parent par T (j) in T . By an inversion of T ∈ SPT(G), we mean an ordered pair (i, j) of vertices such that i is an ancestor of j in T with i > j. The total number of inversions of a spanning tree T is denoted by inv(T ). An inversion (i, j) of T is called a κ-inversion of T if i is not the root r and par T (i) is adjacent to j in G. The κ-number κ(G, T ) of T in G is given by In the definition of G-parking function, we have taken root r = 0. For a root r different from 0, a notion of G-parking functions (with repect to root r) P : V \ {r} −→ N can be easily defined (see [12]). We are now in a position to describe DFS burning algorithm. Let G be a simple graph with a root r ∈ V . Applied to an input function P : V \ {r} −→ N, the DFS algorithm of Perkinsons-Yang-Yu [12] gives a subset burnt vertices of burnt vertices and a subset tree edges of tree edges as an output. We imagine that a fire starts at the root r and spread to other vertices of G according to the depth-first rule. The value P(j) of the input function P can be considered as the number of water droplets available at vertex j that prevents spread of fire to j. If i is a burnt vertex, then consider the largest non-burnt vertex j adjacent to i. If P(j) = 0, then fire from i will spread to j. In this case, add j in burnt vertices and include the edge (i, j) in tree edges. Now the fire spreads from the burnt vertex j. On the other hand, if P(j) > 0, then one water droplet available at j will be used to prevent fire from reaching j through the edge (i, j). In this case, the dampened edge (i, j) is removed from G, number of water droplets available at j is reduced to P(j) − 1 and the fire continue to spread from the burnt vertex i through non-dampened edges. If all the edges from i to unburnt vertices get dampened, then the search backtracks. At the start, burnt vertices = {r} and tree edges = {}.
Perkinson, Yang and Yu [12] constructed a bijection φ : PF(G) −→ SPT(G) using their DFS algorithm. We state their result for future reference.
Theorem 2.8 (Perkinson-Yang-Yu). Let G be a simple graph on V with root r. If on applying DFS burning algorithm to P : V \ {r} −→ N, the subset burnt vertices of burnt vertices is V , then P is a G-parking function and the tree edges of tree edges form a spanning tree φ(P) of G. Further, the mapping P → φ(P) given by DFS algorithm induces a bijection φ : PF(G) −→ SPT(G) such that rsum(P) = g(G) − sum(P) = κ(G, φ(P)).
Let P∈PF(G) q rsum(P) be the reversed sum enumerator for G-parking functions. Then Theorem 2.8 establishes the following identity For G = K n+1 with root 0, PF(K n+1 ) = PF(n) is the set of ordinary parking functions of length n and the above identity reduces to the identity P∈PF(n) q rsum(P) = T ∈SPT(K n+1 ) q inv(T ) proved by Kreweras [6].
As g(K n ) = n 2 − n + 1, P(i) = P(i) − 1 and P(r) = 0, we have rsum( P ) = n 2 − sum(P) + 1. We now describe the DFS burning algorithm of Gaydarov-Hopkins [5] for multigraphs. Consider a connected multigraph G on V = {0, 1, . . . , n} with root r. Let E(i, j) = E(j, i) be the set of edges between distinct vertices i and j. Fix a total order on E(i, j) for all distinct pair {i, j} of vertices and write E(i, j) = {e 0 ij , e 1 ij , . . . , e Thus we assume that edges of the multigraph G are labelled. Applied to an input function P : V \ {r} −→ N, the DFS algorithm for multigraphs gives a subset burnt vertices of burnt vertices and a subset tree edges of tree edges with nonnegative labels on them as an output. As in the case of DFS algorithm for simple graphs, we imagine that a fire starts at the root r and spread to other vertices of G according to the depth-first rule. If i is a burnt vertex, then consider the largest non-burnt vertex j adjacent to i. If P(j) < a ij = |E(i, j)|, then P(j) edges with higher labels, namely e a ij −1 ij , . . . , e a ij −P(j) ij will get dampened, the edge e a ij −P(j)−1 ij with label a ij − P(j) − 1 will be added to tree edges and j in included in burnt vertices. Now fire will spread from the burnt vertex j. On the other hand, if P(j) ≥ a ij , then all the edges in E(i, j) get dampened and P(j) reduced to P(j) − a ij . The fire continue to spread from the burnt vertex i through non-dampened edges. If all the edges from i to unburnt vertices get dampened, then the search backtracks. At the start, burnt vertices = {r} and tree edges = {}. Gaydarov and Hopkins [5] extended Theorem 2.8 to multigraphs using the DFS burning algorithm for multigraph. We state their result without proof. The reduced spherical K a,b n+1 -parking function P associated to P ∈ sPF(K a,b n+1 ) is given by . In other words, where T = φ b n (P) and r is the root of T .
Proof. Let P ∈ sPF(K a,b n+1 ). Then P ∈ PF(K a,b n+1 ). Choose the largest vertex r of K b n = K a,b n+1 \ {0} such that P(r) < b. We claim that P(j) < b for some j < r. Otherwise, P(i) ≥ a+b, ∀ i ∈ [n]\{r}, a contradiction to P ∈ sPF(K a,b n+1 ). Now consider r to be the root of the complete multigraph K b n on [n]. Then P = P | [n]\{r} is a K b n -parking function. On applying the DFS algorithm (Theorem 2.10), we get φ( P ) ∈ U b n with root r. Set weight ω(r) = P(r). The mapping φ b n : sPF(K a,b n+1 ) −→ U b n given by φ b n (P) = φ( P ) is clearly injective. As |sPF(K a,b n+1 )| = |U b n | = b n (n − 1) n−1 , it follows that φ b n is a bijection. Also, Since rsum(P) = g(K a,b n+1 ) − i∈[n] P(i), we verify that rsum( P ) = rsum(P) + ω(r) + 1. Let G be a connected simple graph on V with root 0. Let m [n] be the generator of M G corresponding to [n]. Then to each spherical G-parking function P, we associate a reduced spherical G-parking function P by x P = x P m  Proof. Proceed as in Theorem 2.9.
In the next section, we shall see that φ G need not be surjective.
where the polynomial g n;n−2 (x) is given in the Remark 2.7. Let K m+1,n be the complete bipartite graph on V = {0, 1, . . . , m} {m + 1, . . . , m + n}. Let K a,b m+1,n be the complete bipartite multigraph on V (defined similar to K a,b n+1 ). More precisely, there are a number of edges in K a,b m+1,n between the root 0 and j, while b number of edges between i and j, where i ∈ {1, . . . , m} and j ∈ {m + 1, . . . , m + n}. Let E be the set of all edges of K m+1,n or K a,b m+1,n through the root 0.
We now derive a combinatorial formula for |sPF(K m+1,n )| = |sPF(K m+1,n − E), using a free (non-minimal) cellular resolution of the monomial ideal M K m+1,n −E is an order monomial ideal in the sense of Postnikov and Shapiro [14], the free complex F * (∆) supported on the order complex ∆ = ∆(Σ m+n ) is a free resolution of I ∆ = M (n−2) K m+1,n −E . Using the cellular resolution F * (∆), the multigraded Hilbert series For 1 ≤ i ≤ m + n, let Γ i be the set of order pairs (s, t) of i-tuples s = (s 1 , . . . , s i ) and : s j < m and t j < n} and set Proposition 3.4. The number of spherical K m+1,n -parking functions is given by Proof. Since the Artinian quotient R has finitely many standard monomials, we have In view of Proposition 3.2, we get the desired formula.
From Proposition 3.4, we clearly have |sPF(K m+1,n )| = |sPF(K n+1,m )|. Further, proceeding as in Proposition 3.4, it can be shown that |sPF(K a,b m+1,n )| = b m+n |sPF(K m+1,n )|. We now compute the number |sPF(G)| of spherical G-parking functions for G = K n+1 − {e}, where e is an edge not through the root 0. We first consider the case e = e 1 , where e 1 = (1, n) is the edge joining 1 and n. As sPF(G) ⊆ PF(G) for G = K n+1 − {e 1 }, on applying Corollary 2.12, we get an injective map φ G : sPF(G) −→ U n , where U n = U G−{0} is the set of uprooted trees on [n] with no edge between 1 and n (i.e., 1 n).
Proof. Let P ∈ sPF(G) and P ∈ PF(G) be the associated reduced spherical G-parking function. Choose the largest vertex r of G − {0} such that P(r) = 0. We claim that P(j) = 0 for some j < r. Otherwise, P(i) ≥ 2, ∀ i ∈ [n] \ {r}, a contradiction to P ∈ sPF(G). Now consider r to be the root of the graph G = G − {0} on [n]. Then P = P | [n]\{r} is a G -parking function. On applying Theorem 2.8, we get a spanning tree φ( P ) of the graph G with root r. Since P (i) ≥ 1 for i > r, all the edges (r, i) are dampened. Hence, φ( P ) is a uprooted spanning tree of G with root r. Define φ G (P) = φ( P ). Clearly, φ G : sPF(G) −→ U n is injective.
We now show that φ G is surjective. Let T ∈ U n with root r. From Theorem 2.9, the map φ n : sPF(K n+1 ) −→ U n is bijective. Thus there exists P ∈ sPF(K n+1 ) such that φ n (P) = φ( P) = T , where P is the reduced spherical parking function associated to P and P = P | [n]\{r} . Claim : P ∈ sPF(G). Let . We shall assume that 1 ∈ A but n / ∈ A. The other case, n ∈ A but 1 / ∈ A is similar. Suppose A = {i 1 , i 2 , . . . , i t } such that 1 = i 1 < i 2 < . . . < i t < n. As m A = ( j∈A x j ) n−t+1 and m A = x 1 m A x P , we have P(1) = n − t and P(i k ) ≥ n − t + 1 for k = 2, . . . , t. Let [n] \ A = {r = j 1 , j 2 , . . . , j s } such that P(j 1 ) ≤ . . . ≤ P(j s ). Then s + t = n. Since P ∈ sPF(K n+1 ), we must have P(j 1 ) = 1, P(j 2 ) < 2, . . . , P(j s ) < s. This shows that P(j 2 ) = 0, . . . , P(j s ) < s − 1, P(1) = s − 1 and P(i k ) ≥ s, for 2 ≤ k ≤ t. Now we apply DFS algorithm to get spanning tree φ( P) with root r. Starting from the root r, all the vertices j 2 , . . . , j s get burnt in the first s − 1 steps. Also, whenever certain edges joining j l and i k get dampened, it reduces the value P(i k ) by 1 for k = 1. Since j l = n for some l, after the (s − 1) th step of DFS algorithm, P(1) = s − 1 and the reduced values of P(i k ) are all ≥ 1 for 2 ≤ k ≤ t. The value P(1) reduces by at most 1 if the search backtracks from j l to j l−1 for l ∈ {2, . . . , s}. Again, as j l = n for some l and 1 n, we see that the reduced value of P(1) is 1 even after the search backtracks to the root r = j 1 . Hence, 1 is not a burnt vertex, a contradiction to φ( P) = T . This proves the claim and the theorem. (1) By renumbering vertices of G, we easily see that for any edge e between distinct vertices i, j ∈ [n]. (2) Let e = (n − 1, n) be the edge in K n+1 joining n − 1 and n and G = K n+1 − {e}. For n ≥ 3, the injective map φ G : sPF(G) −→ U G−{0} need not be surjective. In fact, for n = 4, |sPF(K 4+1 − {e})| = 12 but the number of uprooted trees on [4] with no edge between 3 and 4 is exactly 17.
1n , e 1 1n , . . . , e b−1 1n } be the set of all edges joining 1 and n in the complete multigraph K a,b n+1 . We recall that U b n is the set of uprooted trees T on [n] with a label : E(T ) → {0, 1, . . . , b − 1} on its edges and a weight ω(r) ∈ {0, 1, . . . , b − 1} on its root r. Let U b n = {T ∈ U b n : 1 n in T}. Then using Theorem 2.11, the bijection of the Theorem 3.5 can be extended to a bijection In particular, |sPF(K a,b We now determine the number |U n | of uprooted trees on [n] with 1 n. Let T n,0 be the set of labelled trees on [n] such that the root has no child (or son) with smaller labels. Let A n be the set of labelled rooted-trees on [n] with a non-rooted leaf n. Chauve, Dulucq and Guibert [1] constructed a bijection η : T n,0 → A n . As earlier, let U n be the set of uprooted trees on [n]. Also, let B n be the set of labelled rooted-trees on [n] with a non-rooted leaf 1. We see that there are bijections U n → T n,0 and B n → A n obtained by simply changing label i to n − i + 1 for all i. The bijection η : T n,0 → A n induces a bijection ψ : U n → B n . For sake of completeness, we briefly describe construction of the bijection ψ essentially as in [1].
Let T ∈ U n with root r. Note that r = 1.
Step (1) : Consider a maximal increasing subtree T 0 of T containing 1. Let T 1 , . . . , T l be the subtrees (with at least one edge) of T obtained by deleting edges in T 0 . Let r i be the root of T i for 1 ≤ i ≤ l. The root r of T must be a root of one of the subtrees T i . Let r j = r. Then 1 is a leaf of T j .
Step (2) : If T 0 has m vertices, then T 0 is determined by an increasing tree T 0 on [m] and a set S 0 of labels on T 0 . We write T 0 = (T 0 , S 0 ).
Step (3) : Let S 0 = (S 0 \ {1}) ∪ {r}. Then (T 0 , S 0 ) determines an increasing subtree T 0 with root r = min{S 0 }. Graft T j on the increasing subtree T 0 at the root r and obtain a tree T j . Now graft T i (i = j) on T j at r i and obtain a tree T with root r . Also note that 1 is a non-rooted leaf of T .
All the above steps can be reversed, thus ψ(T ) = T defines a bijection ψ : U n → B n .
For n ≥ 3, let U n = {T ∈ U : 1 n in T }. We shall determine the image ψ(U n ) ⊆ B n of U n under the bijection ψ : U n → B n . Let B n = {T ∈ B n : 1 n in T }. Set If T ∈ B , then the unique T ∈ U n with ψ(T ) = T must have 1 n in T , that is, T ∈ U n . Now we consider the remaining case. Let T ∈ B n with root(T ) = r = n and r ∼ n in T . We shall show that ψ(T ) = T for T ∈ U n if and only if 1 is a descendent of n in T (or equivalently, T ∈ B ). Consider the maximal increasing subtree T 0 of T containing the root r . If 1 is a descendent of a leaf r j of T 0 , then the maximal increasing subtree T 0 of T containing 1 is obtained by replacing r j with 1 in the vertex set of T 0 and labeling it as indicated in Step (2) of the construction of ψ. Clearly, r j = r is the root of T . If r j = r = n, then 1 ∼ n in T as r ∼ n in T . Thus, if r j = n, i.e., 1 is not a descendent of n in T , then T / ∈ ψ(U n ). On the other hand, if r j = n, i.e., 1 is a descendent of n in T with 1 n, then root(T ) = r = n and 1 n in T . Proposition 3.9. For n ≥ 3, we have |U n | = (n − 1) n−3 (n − 2) 2 .
Let us consider the subset C = {T ∈ B n : root(T ) = r = n} ⊆ B n . Clearly, B = B B ⊆ C. The enumeration of C is similar to that of A, except now the root r ∈ {2, . . . , n − 1} can take any one of the n − 2 values. Thus |C| = (n − 1) n−3 (n − 2) 2 . We can easily construct a bijective correspondence between A and C \ B. Let T ∈ A. Then 1 n in T and root(T ) = n. Consider the unique path from the root n to the leaf 1 in T . As 1 n in T , the childr of n lying on this unique path is different from 1. LetT be rooted tree consisting of the tree T with the new rootr. As root(T ) =r = n,r ∼ n and 1 is not a descendent of n inT , we haveT ∈ C \ B. The mapping T →T from A to C \ B is clearly a bijection. IfT ∈ C \ B, then root(T ) =r = n,r ∼ n and 1 is not a descendent of n inT . Now unique T ∈ A that maps toT is the rooted tree obtained from T by taking n as the new root. Thus |A| = |C \ B| and hence, |U n | = |C| = (n − 1) n−3 (n − 2) 2 . Proof. In view of Theorem 3.5 and Remarks 3.6, the result follows.