Physlib.QFT.PerturbationTheory.WickContraction.Card

For a set of $n+1$ elements, the total number of Wick contractions is equal to the sum of the number of Wick contractions where the element $0$ is not paired with any other element and the number of Wick contractions where the element $0$ is paired with some element. That is, $$|\text{WickContraction}(n+1)| = |\{c \in \text{WickContraction}(n+1) \mid 0 \text{ is unpaired in } c\}| + |\{c \in \text{WickContraction}(n+1) \mid 0 \text{ is paired in } c\}|$$ where the pairing status of element $0$ in a contraction $c$ is determined by whether the dual of $0$ exists (i.e., `(c.getDual? 0).isSome`).

Let $\text{WickContraction}(k)$ denote the set of Wick contractions on $k$ elements. For any natural number $n$, the number of Wick contractions on $n+1$ elements where the element at index 0 is uncontracted (i.e., not paired with any other element) is equal to the total number of Wick contractions on $n$ elements.

Let $W_{n+1}$ be the set of Wick contractions on $n+1$ indices $\{0, 1, \dots, n\}$. For a contraction $c \in W_{n+1}$, let $c(0)$ denote the index paired with index $0$. The number of Wick contractions where $0$ is paired with some other index is equal to the sum over all possible indices $j \in \{1, \dots, n\}$ of the number of Wick contractions where index $0$ is paired specifically with $j$: $$|\{ c \in W_{n+1} \mid c(0) \text{ is some index} \}| = \sum_{i=0}^{n-1} |\{ c \in W_{n+1} \mid c(0) = i+1 \}|$$ where $i$ ranges over the finite set of $n$ elements.

Given a natural number $n$, a finite set $a \subseteq \{0, \dots, n-1\}$, and an index $i \in \{0, \dots, n\}$, let $f_i: \{0, \dots, n-1\} \to \{0, \dots, n\}$ be the embedding that skips the value $i$ (defined by $f_i(j) = j$ if $j < i$ and $f_i(j) = j+1$ if $j \ge i$). Let $\sigma: \{0, \dots, n\} \to \{1, \dots, n+1\}$ be the successor map $\sigma(x) = x+1$. Then the image of the set $a$ under the composition $\sigma \circ f_i$ is disjoint from the set $\{0, i+1\}$.

For a Wick contraction $c$ on $n$ elements and an index $i \in \{0, \dots, n\}$, the function constructs a new Wick contraction on $n+2$ elements (indices $\{0, \dots, n+1\}$). The resulting contraction is formed by taking the existing pairs in $c$, relabeling their indices to the set $\{1, \dots, n+1\} \setminus \{i+1\}$, and adding the new contraction pair $\{0, i+1\}$.

Let $n$ be a natural number and $c$ be a Wick contraction on $n$ elements. For any index $i \in \{0, \dots, n\}$, let $c'$ be the Wick contraction on $n+2$ elements (indices $\{0, \dots, n+1\}$) constructed by adding the contraction pair $\{0, i+1\}$ and relabeling the existing pairs in $c$. Then, the partner of the index $0$ in $c'$ is $i+1$.

Let $c$ be a Wick contraction on $n$ elements. For any index $i \in \{0, \dots, n\}$, let $c'$ be the Wick contraction on $n+2$ elements formed by adding the pair $\{0, i+1\}$ to $c$ (using the `consAddContract` construction). In this new contraction $c'$, the partner (or dual) of the element $i+1$ is $0$.

Let $c$ be a Wick contraction on $n$ elements $\{0, \dots, n-1\}$. For any index $i \in \{0, \dots, n\}$, let $c'$ be the Wick contraction on $n+2$ elements $\{0, \dots, n+1\}$ constructed by the operation `consAddContract(i, c)`, which adds the contraction pair $\{0, i+1\}$ and relabels the existing pairs in $c$. Let $f_i: \{0, \dots, n-1\} \to \{0, \dots, n\}$ be the embedding that skips the index $i$ (i.e., $f_i(j) = j$ for $j < i$ and $f_i(j) = j+1$ for $j \ge i$), and let $\sigma: \{0, \dots, n\} \to \{1, \dots, n+1\}$ be the successor map $\sigma(x) = x+1$. A subset $a \subseteq \{0, \dots, n-1\}$ is a pair in the contraction $c$ if and only if its image under the composition $(\sigma \circ f_i)$ is a pair in the contraction $c'$.

For any natural number $n$ and any index $i \in \{0, \dots, n\}$, let $\text{consAddContract}_i$ be the operation that takes a Wick contraction $c$ on $n$ elements and constructs a new Wick contraction on $n+2$ elements by adding the contraction pair $\{0, i+1\}$ and relabeling the indices of the original pairs in $c$. This mapping is injective; that is, if $\text{consAddContract}_i(c_1) = \text{consAddContract}_i(c_2)$, then $c_1 = c_2$.

Let $n$ be a natural number and $i \in \{0, \dots, n\}$. Suppose $c$ is a Wick contraction on $n+2$ elements. If the index $0$ is paired with the index $i+1$ in $c$, then there exists a Wick contraction $c'$ on $n$ elements such that $c$ is the result of extending $c'$ by the pair $\{0, i+1\}$ using the `consAddContract` operation.

For any natural number $n$ and any index $i \in \{0, \dots, n\}$, let $W_k$ denote the set of Wick contractions on $k$ elements. The operation $\text{consAddContract}_i$, which extends a Wick contraction $c \in W_n$ by adding the pair $\{0, i+1\}$ and relabeling the remaining indices, is a bijection between $W_n$ and the set of Wick contractions on $n+2$ elements where the index $0$ is paired with the index $i+1$.

Let $W_k$ denote the set of Wick contractions on $k$ elements. For a natural number $n$, the number of Wick contractions on $n+2$ indices where the index $0$ is paired with some other index is equal to $n+1$ times the total number of Wick contractions on $n$ indices: $$|\{ c \in W_{n+2} \mid (c.\text{getDual? } 0).\text{isSome} \}| = (n + 1) \cdot |W_n|$$ where $(c.\text{getDual? } 0).\text{isSome}$ indicates that index $0$ is part of a contraction pair in $c$.

The function $f: \mathbb{N} \to \mathbb{N}$ calculates the number of Wick contractions for $n$ elements. It is defined by the recursive formula: - $f(0) = 1$ - $f(1) = 1$ - $f(n + 2) = f(n + 1) + (n + 1) f(n)$ for any $n \in \mathbb{N}$. This sequence corresponds to the number of involutions on $n$ letters (OEIS:A000085).

For any natural number $n$, the cardinality of the set of Wick contractions on $n$ elements is given by the function $f(n)$, which is defined by the recursive formula: - $f(0) = 1$ - $f(1) = 1$ - $f(n + 2) = f(n + 1) + (n + 1) f(n)$ for any $n \in \mathbb{N}$. This sequence corresponds to the number of involutions on $n$ letters, which is listed as sequence A000085 in the Online Encyclopedia of Integer Sequences (OEIS).