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Let $P([n])$ be the power set of the $n$-element set $[n] := \{1,\dotsc,n\}$. I would like to determine the number of increasing deflationary* maps $f : P([n]) \to P([n])$. These are maps satisfying

  • If $S \subseteq T \subseteq [n]$, then $f(S) \subseteq f(T)$.
  • If $S \subseteq [n]$, then $f(S) \subseteq S$.

For $ n = 0,1,2,3,4$ the numbers are $1, 2, 9, 216, 160000$. I have no result for $n = 5$ so far. The sequence is not on OEIS. However, both the numbers of deflationary maps (A061301) and the numbers of increasing maps (A088322) are on OEIS. Is there a recursive formula? If it is not possible, how can we compute the values for $n \leq 10$?

My approach is as follows. For a subset $\mathcal{A} \subseteq P([n])$ we determine the increasing deflationary maps $\mathcal{A} \to P([n])$ recursively. If $\mathcal{A} \to P([n])$ is given and $T \notin \mathcal{A}$, the extensions to $ \mathcal{A} \cup \{T\} \to P([n])$ correspond to subsets $V \subseteq [n]$ (the image of $T$) such that

  • If $S \in \mathcal{A}$ with $S \subseteq T$, then $f(S) \subseteq V$.
  • We have $V \subseteq T$.

In other words, $V$ is a set between $\bigcup_{S \in \mathcal{A},\, S \subseteq T} f(S)$ and $T$.

Since we only want to build up to $P([n])$, we may also assume that $\mathcal{A}$ is an initial segment of $P([n])$.

With this we can write a program that computes the number of increasing deflationary maps on $P([n])$ for fixed $n$, but it is quite inefficient and does not terminate for $n = 5$ on my machine. A recursive formula would be nice, but the above description cannot be used for this since the first condition really depends on $f$.

Or is there a general way to count the increasing deflationary maps $P \to P$ for a finite partially ordered set $P$? Another example was discussed in SE/4443894.

*deflationary = regressive

Martin Brandenburg
  • 181,922

1 Answers1

1

The answer is $\psi(n-1)^n$, where $\psi(-)$ is the sequence of Dedekind numbers (A000372), extended by $\psi(-1)=0$.

Proof: Let $X$ be a set. There is a bijection between increasing maps $P(X) \to P(X)$ and families $(E_x)_{x \in X}$ of upper sets $E_x \subseteq P(X)$, the corresponding increasing map is $A \mapsto \{x \in X : A \in E_x\}$. It restricts to a bijection between the deflationary increasing maps $P(X) \to P(X)$ and those families $(E_x)_{x \in X}$ such that each $A \in E_x$ contains $x$. Then $E_x$ is just an upper set of $\{A \in P(X) : x \in A\}$, which is order-isomorphic to $P(X \setminus \{x\})$. If $X$ has $n$ elements, there are $\psi(n-1)$ upper sets in $P(X \setminus \{x\})$, so in total there are $\psi(n-1)^n$ deflationary increasing maps $P(X) \to P(X)$. $\checkmark$

The first $10$ Dedekind numbers are known (the last one, $\psi(9)$, has been computed in 2023), and hence the first $11$ values for $\psi(n-1)^n$ are:

1
2
9
216
160000
133827821568
189827435159733571331481
1801744455565479313029125519598327672176553107584
1155791820745758127110805140703797670866815307189659101544198736200639721556530916755391753942016256
5530772904791705705057766136658182789762923249660151131254588630240066119684290297100676713798083901870790632269608400501343270246328654009593415906860566273361339887183518849891113375020463400935978958848
3711304570979402667659964610274483234382771894266788619917520916103531280074096206529189647035905630609036515151562480825996263993047727519075024072512142002522258994358312283680007865846889926104392391369691682588995419988980767525965669355767539976383816740580768760963523724611106887382738228623048615578434404808499713716964809507090654293687785736648151250141716473466884387605263486524410300888559098763723776

Question: Is this sequence interesting enough to be added to OEIS?

Martin Brandenburg
  • 181,922