Discretized configurations and partial partitions

We show that the discretized configuration space of $k$ points in the $n$-simplex is homotopy equivalent to a wedge of spheres of dimension $n-k+1$. This space is homeomorphic to the order complex of the poset of ordered partial partitions of $\{1,...,n+1\}$ with exactly $k$ parts. We compute the exponential generating function for the Euler characteristic of this space in two different ways, thereby obtaining a topological proof of a combinatorial recurrence satisfied by the Stirling numbers of the second kind.

1. Introduction 1.1. Configurations. The discretized configuration space D k (X) was introduced in [1] as a combinatorial model of the classical configuration space of k-tuples of distinct points in a space X. (When k = 2 this has classically been called the "deleted product".) To define D k (X) it is required that X have the structure of a cell complex. In [1], and in the works of several subsequent authors, the space X is a graph, i.e. a finite 1-complex.
To study D k (X) for higher dimensional X, it is natural to begin with some basic building blocks. In this paper we consider the discretized configuration spaces D k (∆ n ), where ∆ n is the n-dimensional simplex. Note that if X is any simplicial complex, then D k (X) is built out of (products of) spaces of the form D i (∆ n ). We prove Theorem 1.1. The space D k (∆ n ) is homotopy equivalent to a wedge of spheres of dimension n − k + 1.

Theorem 1.2. The number of spheres in the wedge is given by the formula
More concisely, the Euler characteristic χ k,n of D k (∆ n ) has the two-variable exponential generating function given by k,n χ k,n−1 x k k! y n n! = e x+y−xe −y .
Thus combinatorial manipulations are sufficient to prove that the two statements in Theorem 1.2 are equivalent.

1.2.
Partitions. The partition lattice Π n is a classical combinatorial object studied since antiquity. It is a poset whose elements are set partitions of [n] = {1, . . . , n} and whose ordering is given by refinement. The order complex of Π n has the homotopy type of a wedge of spheres (see [12]). The (larger) lattice Π ≤n of partial partitions has recently also been shown to have the homotopy type of a wedge of spheres [4]. These results are proved using the machinery of algebraic combinatorics. The standard technique is to find a shelling of the order complex, often via an EL-or CL-labelling of the poset [12]. One nice feature of these arguments is that they often produce an explicit basis for the (unique) nonzero homology group, and one can usually compute its Betti number.
1.3. Connection. The face poset of the space D k (∆ n−1 ) can be identified with a combinatorial object closely related to Π ≤n , namely the posetΠ k ≤n of ordered partial partitions of [n] with exactly k parts. Thus Theorems 1.1 and 1.2 imply the following. Theorem 1.3. The posetΠ k ≤n has the homotopy type of a wedge of spheres of dimension n − k. With notation from Theorem 1.1, the number of spheres is β k,n−1 and the Euler characteristic is χ k,n−1 .
Note that the symmetric group S k acts on the space D k (X) (and on the poset Π k ≤n ) by permuting coordinates. The quotient space U D k (X) has a topological interpretation as an unordered discretized configuration space. The quotient poset Π k ≤n is naturally a subposet of Π ≤n ; its elements are the partial partitions of [n] with exactly k parts. However, Π k ≤n does not have the homotopy type of a wedge of spheres, as can be seen in the case k = 2 (where one obtains a real projective plane RP n−2 ).
Several other authors have studied the topology of the partition lattice and numerous related posets [2,3,5,8,9,10,11]. Many (but not all) of these spaces have the homotopy type of a wedge of spheres. These authors also study the actions of the symmetric group on the homology of the posets. We do not attempt this in the present paper but believe that a careful analysis of the symmetric group action on H n−k+1 (D k (∆ n )) would be interesting.
1.4. Paper contents. We start with definitions, examples, and results in Section 2. In Section 2.3 we show that Theorem 1.3 is equivalent to Theorems 1.1 and 1.2. The proof of Theorem 1.1 is a direct computation using algebraic topology. It takes two steps: in Section 3, we give an inductive argument that the spaces are simply connected (when the dimension is at least 2), and then in Section 4 a spectral sequence computation shows that the spaces have the same homology as a wedge of spheres. Together, these allow us to use the Whitehead theorem to deduce the result.
In particular, we do not give a shelling of the spaces D k (∆ n ), although we suspect that one may be possible.
Problem. Find a combinatorial proof of Theorem 1.1.
In Section 5 we prove Theorem 1.2 by using the interpretation as a partition lattice to count the i-dimensional cells in D k (∆ n ). This count involves the Stirling numbers of the second kind, denoted N K which (by definition) means the number of partitions of a set of size N into exactly K nonempty subsets. The theorem is proved using well-known and elementary facts about Stirling numbers.
We also obtain a recurrence for the top Betti number by following through the spectral sequence. One can prove that the expression for β k,n in Theorem 1.2 agrees with the actual Betti numbers for small k and satisfies the same recurrence; this establishes (the first half of) Theorem 1.2 in a different way.

Definitions, theorem, examples
One can also describe this as the union of those open cells of (∆ n ) k whose closure misses the diagonal; explicitly, We visualize this space by imagining k "robots" in the space ∆ n ; then D k (∆ n ) is the space of allowable configurations if a robot is said to "occupy" the entire closure of the cell in whose interior it is contained, and a configuration is "allowable" if no two robots occupy the same point of ∆ n . The maximum dimension of a cell of D k (∆ n ) is n − k + 1.
Proof. If n − k + 1 = 0, then D k (∆ n ) is a discrete set of k! ≥ 2 points, hence a wedge of 0-spheres. If n − k + 1 = 1, then D k (∆ n ) is a connected 1-complex, hence up to homotopy, a wedge of circles. If n − k + 1 > 1, then 2 ≤ k < n, so Proposition 1 (Section 3) shows that D k (∆ n ) is simply connected and Proposition 2 (Section 4) shows that D k (∆ n ) has the homology of a wedge of spheres of dimension n − k + 1. As there is clearly a map from a wedge of spheres inducing isomorphisms on homology, the Whitehead theorem implies the result.

2.2.
Examples. Let K d denote the complete graph on the vertex set [d] (i.e., the 1-skeleton of ∆ d−1 ). Let S d denote the d-sphere. Let e i denote the ith standard basis vector of R d , for 1 ≤ i ≤ d.
Example 2.1 (k = 2). We view ∆ n as the convex hull of the n + 1 standard basis vectors e i in R n+1 . Then the Gauss map from D 2 (∆ n ) to R n+1 given by (x, y) → x − y is a homeomorphism onto S, the boundary of the polytope with vertex set {e i − e j | i, j ∈ [n + 1], i = j} in R n+1 . To see this, note that the map is obviously continuous and surjective, and given a point in S one can determine the coordinates x and y of the preimage as the "positive" and "negative" parts of S. Thus the map is a bijection.
The complex S is contained in the hyperplane { x i = 0} ∼ = R n and is homeomorphic to S n−1 . This agrees with Theorems 1.1 and 1.2. Note also that D 2 (∆ n−1 ) is contained in D 2 (∆ n ) as an equator.
We next describe some small cases.
is a connected 1-complex with six vertices, each of valence two, and six edges. It is a hexagon. It is the same as D 2 (K 3 ), since neither robot may venture into the interior of the 2-cell of ∆ 2 .  n = 4. The space D 2 (∆ 4 ) is homeomorphic to S 3 . The space D 2 (K 5 ) is a subcomplex which is itself homeomorphic to a closed orientable surface of genus six. This surface is a Heegaard splitting of D 2 (∆ 4 ). The complementary handlebodies are made of prisms (∆ 2 × I and I × ∆ 2 ) and 3-simplices (v × ∆ 3 and ∆ 3 × v).
Example 2.2 (k = n). In this case D k (∆ n ) is a connected 1-complex; a simple count reveals that the rank of H 1 is 1 2 (n − 2)(n + 1)! + 1. It is true but not obvious that this agrees with Theorem 1.2. If one quotients by the action of the symmetric group S k that permutes coordinates, the result is the graph K n+1 (whose first Betti number is n 2 ).

2.3.
Partitions. An ordered partition π of the set [n] = {1, . . . , n} is an r-tuple (for some r) of disjoint nonempty sets π = (S 1 , . . . , S r ) whose union is [n]. (The ordering is on the set of parts, but each part is an unordered set.) An ordered partial partition of [n] is the same, except the union is only required to be a subset of [n]. An ordered partial partition with exactly k parts is an ordered partial partition with r = k.
LetΠ k ≤n be the poset whose elements are the ordered partial partitions of [n] with exactly k parts. The partial order is given by (S 1 , . . . , S k ) ≤ (T 1 , . . . , T k ) iff S i ⊆ T i for each i.
Note thatΠ k ≤n is not a lattice: writing (S 1 , . . . , S k ) as (S i ), the meet (S i )∧(T i ) is (S i ∩T i ) if all these sets are nonempty, but it is otherwise nonexistent; and similarly the join (S i ) ∨ (T i ) is (S i ∪ T i ) if all these sets are disjoint, but otherwise it does not exist. Of course we may add a top and bottom element if we like. For comparison, the poset of partial partitions Π ≤n (see [4]) has a top element consisting of the 1-part partition [n], and it is a lattice provided one includes an empty partition at the bottom.
Proof of Theorem 1.3. The face poset of D k (∆ n−1 ) is isomorphic to the posetΠ k ≤n , by mapping the face σ 1 × · · · × σ k to the element (S i ), where S i is the set of vertices of the cell σ i . Thus the order complex ofΠ k ≤n is homeomorphic to D k (∆ n−1 ), so this is equivalent to Theorems 1.1 and 1.2.
We remark once again that the quotient Π k ≤n ofΠ k ≤n by the symmetric group S k does not have the homotopy type of a wedge of spheres. When k = 2, for example, the action is antipodal and the quotient is a projective plane. Nevertheless Π k ≤n is a subposet of Π ≤n , which is a wedge of spheres up to homotopy (see the introduction).

The fundamental group
This is the first of the propositions referred to in the proof of Theorem 1.1. Proposition 1. If 1 ≤ k < n, then D k (∆ n ) is simply connected.
Proof. If k = 1, then D k (∆ n ) = ∆ n , which is simply connected. If k = 2 we have already seen that D 2 (∆ n ) is homeomorphic to S n−1 , which is simply connected if n > 2. We proceed by induction on k; let k > 2 be fixed. Note that the hypothesis means that if all robots are at vertices, then there are at least two unoccupied vertices.
To prove the theorem we will construct a set of generators of π 1 (D k (∆ n )) and then we will show that each is null-homotopic.
Since D k (∆ n ) ⊂ (∆ n ) k , projection onto the first factor induces a map ρ : D k (∆ n ) −→ ∆ n . The inverse image of a point in the interior of an i-cell of ∆ n is isomorphic to D k−1 (∆ n−i−1 ). In particular, if v is a vertex of ∆ n , then ρ −1 (v) is simply connected, by induction.
Let v be the vertex n + 1 of ∆ n and let T be the spanning tree of the 1-skeleton of ∆ n (that is, K n+1 ) consisting of all edges incident with v. The space ρ −1 (T ) is the union of n + 1 vertex spaces (i.e. the preimages of the vertices), each of which is a copy of the simply connected space D k−1 (∆ n−1 ), and n edge spaces (preimages of edges), each of which is a copy of the connected (but not necessarily simply connected) space I × D k−1 (∆ n−2 ). The edge spaces are attached to the vertex spaces by embeddings at the ends {0, 1} × D k−1 (∆ n−2 ). Thus by the Seifert-van Kampen theorem, ρ −1 (T ) is simply connected. Now consider Y = ρ −1 (K n+1 ). The space Y is obtained from ρ −1 (T ) by attaching n 2 edge spaces I × D k−1 (∆ n−2 ) indexed by the pairs i, j ∈ [n] with i < j. As there are no new vertex spaces, each such edge space results in an HNN extension of the fundamental group; thus π 1 (Y ) is free of rank n 2 .

AARON ABRAMS, DAVID GAY, AND VALERIE HOWER
Note that the entire 1-skeleton of D k (∆ n ) is contained in Y . Thus a generating set for π 1 (Y ) will also generate π 1 (D k (∆ n )). We now describe such a generating set.
Fix a basepoint ⋆ ∈ ρ −1 (v). For each i, j ∈ [n] choose a path α ij in ρ −1 (v) from ⋆ to a configuration x with i and j unoccupied and each robot at a vertex of ∆ n . Let γ ij be the loop starting at x that leaves all robots fixed except the first and moves the first robot around the triangle v → i → j → v. The loop α ij γ ij α −1 ij represents the generator of π 1 (Y ) arising from attaching the edge space ρ −1 ([i, j]). Letting i, j vary, these n 2 loops form a free basis for π 1 (Y ). But clearly each of these generators of π 1 (Y ) is null-homotopic in D k (∆ n ), as the loop γ ij bounds a 2-simplex in D k (∆ n ). We conclude that D k (∆ n ) is simply connected, as desired.

Homology
Here we prove the second of the propositions referred to in the proof of Theorem 1.1, using a spectral sequence to compute the homology of D k (∆ n ). We refer the reader to [6] for a discussion on the use of spectral sequences in combinatorics. In what follows, all homology groups will have integer coefficients.
Recall or observe: is k! points. Let n ≥ 1 and 1 ≤ k ≤ n be fixed. Again we consider the projection ρ : D k (∆ n ) −→ ∆ n onto the first coordinate. Note that ρ satisfies where i 1 , i 2 , i 3 , . . . , i n−k+2 are distinct vertices of ∆ n and the face [·] of ∆ n is the interior of the convex hull of the given vertices. We use ∆ n (k) to denote the kdimensional faces of ∆ n and ∆ n ≤k = i≤k ∆ n (i) for the k-skeleton of ∆ n . The map ρ gives a filtration of D k (∆ n ) as follows:

DISCRETIZED CONFIGURATIONS AND PARTIAL PARTITIONS 7
We can hence construct a spectral sequence [7, p. 327] (E r , d r ) with ) converging to homology with closed supports. Since H p (R p ) = Z is the only nonzero homology group of R p , we have where we have used the Künneth formula.
Proposition 2. Let n ≥ 1 and 1 ≤ k ≤ n. Then and H n−k+1 (D k (∆ n )) is free abelian and nontrivial. Thus D k (∆ n ) has the same homology as a wedge of spheres of dimension n − k + 1.
Proof. We induct on k; the cases k = 1, 2 are observations (2), (3) above. Assume the theorem holds for configurations of k − 1 robots. Let (E r , d r ) be the spectral sequence from above. Using our inductive hypothesis, the E 1 term has nonzero entries only along the diagonal line p + q = n − k + 1 and along row q = 0. The entries in the E 1 term are as follows: and all other entries are zero.
The only possible nonzero higher differentials are the horizontal maps d 1 p,0 : is a direct sum of maps Note that if dimf ≤ n − k and g ∈ f (p − 1), then the map is injective (hence an isomorphism) as it is induced by inclusion. Thus, computing E 2 p,0 = kerd 1 p,0/imd 1 p+1,0 is equivalent to computing the pth homology group of ∆ n for p < n − k, and we obtain E 2 p,0 = 0 for p < n − k. For p = n−k, we have that d 1 n−k,0 is equivalent to ∂ n−k , where ∂ is the boundary map for the n-simplex. For each (n − k + 1)-face f of ∆ n , let γ f be a generator for the factor of H 0 (D k−1 (∆ k−2 )) corresponding to the fiber over f . Then and hence imd 1 n−k+1,0 ∼ = im∂ n−k+1 = ker∂ n−k ∼ = kerd 1 n−k,0 , which yields E 2 n−k,0 = 0. Thus the only nonzero entries of the E 2 term are E 2 0,0 ∼ = Z and also along the line p + q = n − k + 1 where we have free abelian groups. We hence obtain the integer homology groups of D k (∆ n ) from E 2 = E ∞ by adding along the lines p + q = r, which proves the inductive step.

The Euler characteristic
In this section we discuss two proofs of Theorem 1.2. We give the first in detail, via the interpretation of D k (∆ n ) as the order complex of the posetΠ k ≤n+1 . The second approach uses the spectral sequence computation from Section 4; we give an outline and invite the reader to fill in the details.

Stirling numbers.
Recall that the symmetric group S k acts on D k (∆ n ) by permuting coordinates. This is a free cellular action; i.e., the quotient U D k (∆ n ) inherits a natural cell structure. An i-dimensional cell of D k (∆ n ) corresponds to an ordered partial partition of [n + 1] which uses exactly k + i of the elements from [n + 1], and an i-dimensional cell of U D k (∆ n ) corresponds to an unordered partial partition of [n + 1] which uses exactly k + i of the elements from [n + 1].
The Stirling number of the second kind, denoted N K , is by definition the number of ways to partition a set of size N into exactly K nonempty subsets. (These partitions are unordered.) Thus the number of i-cells of U D k (∆ n ) is and the Euler characteristic of U D k (∆ n ) is Note that this sum can be extended to all integer values of i, since the additional terms would all be zero for one reason or another. As the Euler characteristic is multiplicative under covers, it follows that the Euler characteristic of D k (∆ n ) is (5.1) χ k,n = k! i (−1) i n + 1 k + i k + i k .