InterlacingNumber

Pairs of blocks whose i-th arcs interlace, each block closed by an arc to infinity.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt001781via SetPartition✓↗

InterlacingNumber(partition)Pairs of blocks whose i-th arcs interlace, each block closed by an arc to infinity.

Typed signatures

Type	Package	Applies
(set_partition) -> number	combinatorics	always

Domain: Set partition statistics

Details

• Defined over SetPartition as an expression in _x, evaluated by compute-engine — the definition IS the implementation.
• Takes a SetPartition and nothing else — it reads values against their positions, or walks the orbits, so it needs the bijection. Applying it to a bare list is a type error, not a wrong answer.

Definition

Epsil, as the implementation runs it.

on SetPartition, over _x

0 if Length(_x) < 2 else Apply((il_n: integer) => Sum(Map(
                                                           Function(Apply((il_s: list<integer>) => Sum(Map(
                                                                                                            Function(Apply((il_t: list<integer>) => Count(Filter(
                                                                                                                                                                  Range(1, Min(Length(il_s), Length(il_t))),
                                                                                                                                                                  Function((il_n + 1 if i > Length(il_s) else il_s[i]) <
                                                                                                                                                                           (il_n + 1 if i > Length(il_t) else il_t[i]) &&
                                                                                                                                                                           (il_n + 1 if i > Length(il_t) else il_t[i]) <
                                                                                                                                                                           (il_n + 1 if i + 1 >
                                                                                                                                                                                        Length(il_s) else il_s[i + 1]) &&
                                                                                                                                                                           (il_n + 1 if i + 1 >
                                                                                                                                                                                        Length(il_s) else il_s[i + 1]) <
                                                                                                                                                                           (il_n + 1 if i + 1 >
                                                                                                                                                                                        Length(il_t) else il_t[i + 1]) ||
                                                                                                                                                                           (il_n + 1 if i > Length(il_t) else il_t[i]) <
                                                                                                                                                                           (il_n + 1 if i > Length(il_s) else il_s[i]) &&
                                                                                                                                                                           (il_n + 1 if i > Length(il_s) else il_s[i]) <
                                                                                                                                                                           (il_n + 1 if i + 1 >
                                                                                                                                                                                        Length(il_t) else il_t[i + 1]) &&
                                                                                                                                                                           (il_n + 1 if i + 1 >
                                                                                                                                                                                        Length(il_t) else il_t[i + 1]) <
                                                                                                                                                                           (il_n + 1 if i + 1 >
                                                                                                                                                                                        Length(il_s) else il_s[i + 1]), i),
                                                                                                                                                                )), _x[t]), t),
                                                                                                            Range(s + 1, Length(_x)),
                                                                                                          )), _x[s]), s),
                                                           Range(1, Length(_x) -
                                                                    1),
                                                         )), Length(Flatten(_x)))

Examples

Basic #

the partition $\{1,3\} \mid \{2\}$

live ↗

In$\mathrm{InterlacingNumber}({\operatorname{SetPartition}}([[1, 3], [2]]))$Out$1$

the partition $\{1\} \mid \{2,4\} \mid \{3\}$

live ↗

In$\mathrm{InterlacingNumber}({\operatorname{SetPartition}}([[1], [2, 4], [3]]))$Out$1$

Properties #

FindStat's value for {{1}}

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{SetPartition}}([[1]]), \text{InterlacingNumber})$Out$0$

Known value · FindStat St001781

FindStat's value for {{1,2}}

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{SetPartition}}([[1, 2]]), \text{InterlacingNumber})$Out$0$

Known value · FindStat St001781

FindStat's value for {{1,2,3}}

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{SetPartition}}([[1, 2, 3]]), \text{InterlacingNumber})$Out$0$

Known value · FindStat St001781

About

Examples	5
Crosswalk	FindStat