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The Subset Lattice

When X is a finite set, the family of all subsets of X, partially ordered by inclusion, forms a subset latticeA lattice is a special type of poset.

The Subset Lattice

When \(X\) is a finite set, the family of all subsets of \(X\), partially ordered by inclusion, forms a subset latticeA lattice is a special type of poset. You do not have to concern yourself with the definition and can safely replace lattice with poset as you read this chapter.. We illustrate this in where we show the lattice of all subsets of \(\{1,2,3,4\}\). In this figure, note that we are representing sets by bit strings, and we have further abbreviated the notation by writing strings without commas and parentheses.

For a positive integer \(t\), we let \(\bftwo^t\) denote the subset lattice consisting of all subsets of \(\{1,2,\dots,t\}\) ordered by inclusion. Some elementary properties of this poset are:

The height is \(t+1\) and all maximal chains have exactly \(t+1\) points.
The size of the poset \(\bftwo^t\) is \(2^t\) and the elements are partitioned into ranks (antichains) \(A_0, A_1,\dots, A_t\) with \(|A_i|=\binom{t}{i}\) for each \(i=0,1,\dots,t\).
The maximum size of a rank in the subset lattice occurs in the middle, if \(s=\lfloor t/2\rfloor\), then the largest binomial coefficient in the sequence \(\binom{t}{0}, \binom{t}{1},\binom{t}{2},\dots,\binom{t}{t}\) is \(\binom{t}{s}\). Note that when \(t\) is odd, there are two ranks of maximum size, but when \(t\) is even, there is only one.

Sperner's Theorem

For the width of the subset lattice, we have the following classic result of Sperner.

Theorem: Sperner's Theorem

For each \(t\ge1\), the width of the subset lattice \(\bftwo^t\) is the maximum size of a rank, , \[\begin{aligned}\end{aligned}\].

Proof

The width of the poset \(\bftwo^t\) is at least \(C(t,\lfloor\frac{t}{2}\rfloor)\) since the set of all \(\lfloor\frac{t}{2}\rfloor\)-element subsets of \(\{1,2,\dots,t\}\) is an antichain. We now show that the width of \(\bftwo^t\) is at most \(C(t,\lfloor\frac{t}{2}\rfloor)\).

Let \(w\) be the width of \(\bftwo^t\) and let \(\{S_1,S_2,\dots, S_w\}\) be an antichain of size \(w\) in this poset, , each \(S_i\) is a subset of \(\{1,2,\dots,t\}\) and if \(1\le i\lt j\le w\), then \(S_i\nsubseteq S_j\) and \(S_j\nsubseteq S_i\).

For each \(i\), consider the set \(\cgS_i\) of all maximal chains which pass through \(S_i\). It is easy to see that if \(|S_i|=k_i\), then \(|\cgS_i|=k_i!(t-k_i)!\). This follows from the observation that to form such a maximum chain beginning with \(S_i\) as an intermediate point, you delete the elements of \(S_i\) one at a time to form the sets of the lower part of the chain. Also, to form the upper part of the chain, you add the elements not in \(S_i\) one at a time.

Note further that if \(1\le i \lt j\le w\), then \(\cgS_i\cap \cgS_j =\emptyset\), for if there was a maximum chain belonging to both \(\cgS_i\) and \(\cgS_j\), then it would imply that one of \(S_i\) and \(S_j\) is a subset of the other.

Altogether, there are exactly \(t!\) maximum chains in \(\bftwo^t\). This implies that \[\begin{aligned}\end{aligned}\]. This implies that \[\begin{aligned}\end{aligned}\] It follows that \[\begin{aligned}\end{aligned}\]. Thus \[\begin{aligned}\end{aligned}\].

आता तुम्ही हे गणित कोणतेही कॅल्क्युलेटर सोडवू शकत नाही, पण त्याचे भाग गणितीय आहेत. खालीलपैकी एकचा प्रयत्न करा किंवा स्वतःचे टाइप करा.

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मोफत खाते प्रत्येक शिकवणी वर टिपण्णी जोडते, तुम्ही पूर्ण केलेल्या गोष्टीचा नोंदवही, तुमच्या सोडविलेल्या समस्या एका ठिकाणी, आणि या पानाच्या विषयी तुम्ही विचारू शकता अशा शिक्षकाला. गणित स्वतःच सर्वांसाठी खुले आहे, दाखलन झालेले असो किंवा नसो.

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लोक विचारतात ते प्रश्न

Permutation or combination?

Ask whether order matters. A lock code is a permutation (order matters); a hand of cards is a combination (it does not).

What is a graph in this sense?

Dots (vertices) joined by lines (edges), not a plot. Road maps, social networks and molecules are graphs; questions like "is there a route" and "how few colours" are graph theory.

या योजनेचे मुख्य उद्दिष्ट म्हणजे. Keller & Trotter, Applied Combinatorics (CC BY-SA 4.0). नंतर ते पुन्हा स्थिरावले आणि तेथेच ते स्थिरावले; म्हणजेच तेथेच ते स्थिरावले.

अधिक माहिती Combinatorics & Graph Theory

The counting principles Pigeonhole principle and inclusion, exclusion Binomial coefficients and Pascal's triangle Recurrences and generating functions Graphs: vertices, edges, degrees Paths, cycles, trees, Euler and Hamilton Colouring and planar graphs