The Complete Systematic breakdown of - Unix -aka- Linux - PermissionModes

The complete, systematic breakdown of Unix/Linux permission modes requires enumerating every valid combination, and grouping them into mutually exclusive, collectively exhaustive categories — across all axes:

• Axis 1: File type (1st char)
• Axis 2: Owner / Group / Others permission triplets
• Axis 3: Permission levels (r/w/x combinations)
• Axis 4: Special bits (setuid, setgid, sticky)
• Axis 5: Numerical representation (octal)

Below is the full, formal structure. We will proceed layer by layer — from simplest to full combination space.

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🧭 LEVEL 1 — File Type (Position 1)

Symbol	Type	Description	Typical Example
-	Regular file	Standard file	-rw-r--r--
d	Directory	Folder	drwxr-xr-x
l	Symbolic link	Pointer to another file	lrwxrwxrwx
c	Character device	Stream I/O device	/dev/tty
b	Block device	Block I/O device	/dev/sda
p	FIFO / named pipe	IPC channel	/tmp/fifo_pipe
s	Socket	IPC socket	/var/run/docker.sock

→ 7 exhaustive file-type classes, mutually exclusive.

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🧩 LEVEL 2 — Permission Triplets

Each of the next 9 bits (positions 2–10) describe permissions:

• Owner (u) = bits 2–4
• Group (g) = bits 5–7
• Others (o) = bits 8–10

Each triplet can take one of 8 possible values (2³ = 8):

Triplet	r	w	x	Binary	Octal	Meaning
---	0	0	0	000	0	no permissions
--x	0	0	1	001	1	execute only
-w-	0	1	0	010	2	write only
-wx	0	1	1	011	3	write + execute
r--	1	0	0	100	4	read only
r-x	1	0	1	101	5	read + execute
rw-	1	1	0	110	6	read + write
rwx	1	1	1	111	7	full access

→ Each permission triplet = 8 possibilities.

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🔢 LEVEL 3 — Total Basic Permission Space

Since we have 3 triplets (user, group, others):

$$8^3 = 512 \text{ unique permission combinations}$$

Each combination corresponds to an octal triplet (XYZ):

• X = owner permissions (0–7)
• Y = group permissions (0–7)
• Z = others permissions (0–7)

Example mappings:

Symbolic	Octal	Binary (9 bits)	Meaning
rwxr-xr-x	755	111101101	Executable for all, modifiable by owner
rw-r--r--	644	110100100	Readable by all, writable by owner
rwx------	700	111000000	Private executable
rw-------	600	110000000	Private file
rwxrwxrwx	777	111111111	Everyone full access (dangerous)
----------	000	000000000	Completely inaccessible

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🧮 LEVEL 4 — Full 4-Digit Octal Representation (Special Bits Included)

We now add the special bits (the first octal digit):

Bit	Octal Value	Symbolic Indicator	Scope	Meaning
setuid	4	s in user exec position	user	run as file owner
setgid	2	s in group exec position	group	run as file’s group / inherited group
sticky	1	t in others exec position	others	only file owner can delete in shared dir

These three bits form the most significant octal digit, ranging from 0–7:

High Digit	Meaning
0	No special bits
1	Sticky bit only
2	Setgid only
3	Setgid + sticky
4	Setuid only
5	Setuid + sticky
6	Setuid + setgid
7	All three set

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🚩 LEVEL 4 Summary: Full Octal Mode Space

Octal Form	Composition	Example Symbolic	Meaning
0XYZ	No special bits	-rwxr-xr-x	Normal file/dir
1XYZ	Sticky bit	drwxrwxrwt	Shared dir like /tmp
2XYZ	Setgid	rwxr-sr-x	Group inheritance
3XYZ	Setgid + Sticky	rwxr-sr-t	Sticky + group inheritance
4XYZ	Setuid	rwsr-xr-x	Execute as owner
5XYZ	Setuid + Sticky	rwsr-xr-t	Owner exec + sticky
6XYZ	Setuid + Setgid	rwsr-sr-x	Owner + group elevation
7XYZ	All special bits	rwsr-sr-t	Full privilege/sticky

Each XYZ expands to 512 combinations, so full space:

$$8 \times 512 = 4096 \text{ total possible permission combinations}$$

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🧱 LEVEL 5 — Semantic Groupings (Logical Classes)

To make 4096 manageable, we can classify symbolically and semantically:

A. Visibility Control

Category	Definition	Examples (Octal)
Private	Only owner can access	700, 600
Shared Group	Group can read/write	770, 660
Public Read	Anyone can read	755, 644
Public Write	Anyone can write	777 (insecure)
No Access	No one can read/write	000

B. Execution Control

Category	Definition	Example
Executable	At least one x bit set	755, 775, 711
Non-executable	No x bits	644, 600, 664

C. Special Behavior

Category	Effect	Example
setuid	Execute as owner	/usr/bin/passwd (4755)
setgid	Execute as group / inherited group	/usr/bin/mail (2755)
sticky	Only owner can delete	/tmp (1777)

D. Safety/Exposure

Level	Description	Example
Secure	Restricted to owner	600, 700
Controlled	Group-shared	660, 770
Exposed	Public readable	644, 755
Dangerous	Public writable	666, 777

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🧩 LEVEL 6 — Expanded Enumeration by Octal Category

Owner (first triplet / X digit)

X	Owner Permissions	Symbolic	Binary
0	---	none	000
1	--x	execute	001
2	-w-	write only	010
3	-wx	write + exec	011
4	r--	read only	100
5	r-x	read + exec	101
6	rw-	read + write	110
7	rwx	full	111

Same table applies to Group (Y) and Others (Z).

→ Cross product: ( 8 \times 8 \times 8 = 512 ).

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🧰 LEVEL 7 — Practical Canonical Sets

Category	Octal	Symbolic	Use Case
Private files	600	rw-------	SSH keys, config
Private dirs	700	rwx------	Home directories
Group collaboration	660	rw-rw----	Shared writable files
Shared dir	770	rwxrwx---	Group projects
Public readable	644	rw-r--r--	Web docs
Public executable	755	rwxr-xr-x	Scripts, binaries
Sticky shared dir	1777	drwxrwxrwt	/tmp
Setuid binary	4755	rwsr-xr-x	passwd, sudo
Setgid directory	2755	rwxr-sr-x	Shared team folder

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🧮 LEVEL 8 — Symbolic Manipulation (chmod algebra)

Symbolic operators in chmod:

Operator	Meaning
+	Add permission
-	Remove permission
=	Set exactly

Example rules:

• chmod u+x file → add execute for owner
• chmod go-w file → remove write for group/others
• chmod a=r file → set everyone to read-only

Shorthand classes:

Class	Scope
u	user (owner)
g	group
o	others
a	all (u+g+o)

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🔐 LEVEL 9 — Interdependencies (Completeness Map)

Dimension	Values	Mutually Exclusive?	Collectively Exhaustive?
File type	7	✅	✅
Permission triplet	8	✅	✅
Scope (u/g/o)	3	✅	✅
Special bits	8	✅	✅
Octal representation	0000–7777	✅	✅

So the Cartesian product space of possible permission states is:

$$7 \text{(types)} \times 8 \text{(special bits)} \times 512 \text{(permissions)} = 28,672 \text{ possible symbolic states}$$

That’s the complete universe of Unix file mode strings.

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🧠 LEVEL 10 — Conceptual Summary Tree (Taxonomy)

Unix Mode (10-char)
├── File Type (7)
│   ├── Regular (-)
│   ├── Directory (d)
│   ├── Symlink (l)
│   ├── Char Dev (c)
│   ├── Block Dev (b)
│   ├── FIFO (p)
│   └── Socket (s)
├── Permissions (512 = 8³)
│   ├── User (8)
│   ├── Group (8)
│   └── Others (8)
└── Special Bits (8)
    ├── None (0)
    ├── Sticky (1)
    ├── Setgid (2)
    ├── Setgid+Sticky (3)
    ├── Setuid (4)
    ├── Setuid+Sticky (5)
    ├── Setuid+Setgid (6)
    └── All (7)

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✅ TL;DR — The Exhaustive Map

Dimension	Possibilities	Description
File Types	7	- d l c b p s
Permission Triplets per Entity	8 each	--- --x -w- -wx r-- r-x rw- rwx
Entities	3	user, group, others
Special Bits	8	0–7 combinations of sticky/setuid/setgid
Full Permission Octals	0000–7777	4096 unique permission sets
Total Symbolic Mode Strings (with file types)	28,672	the complete universe

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S👀 ALSO: "FULL TABLE"

SPREADSHEET-LINK: Unix_Permission_Combinations_as-a-Combinatory-Table

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What you’re seeing in the spreadsheet for 4-digit octal's, (like 6667 in the sheet) is completely valid but not what’s usually shown by ls -l. Here’s why that happens, and how to interpret it correctly.

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🧮 1. Octal Permissions Have Two Layers

Form	Digits	Meaning	Example	Typical Use
3-digit	XYZ	Base permissions only (owner, group, others)	755 → rwxr-xr-x	Normal chmod/ls
4-digit	SXYZ	Adds the special bits (setuid, setgid, sticky) in the leading digit	4755 → rwsr-xr-x	Complete representation

So the table’s 4-digit octals (e.g., 6667) are full 12-bit modes, not just the 9-bit permission triplets.

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🧩 2. The Composition Explained

The 12-bit mode looks like this internally:

Bit Range	Octal Digit	Role
12–10	S	special bits (setuid/setgid/sticky)
9–7	X	owner permissions
6–4	Y	group permissions
3–1	Z	others permissions

So:

Mode = (SpecialBits × 512) + (Owner × 64) + (Group × 8) + (Others)

In octal shorthand:

SXYZ

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⚙️ 3. Example Breakdown: 6667

Component	Octal	Binary	Meaning
Special bits	6	110	setuid + setgid
Owner	6	110	read + write
Group	6	110	read + write
Others	7	111	read + write + execute
→	Symbolic: rwSrwSrwx	execute under user/group privileges

Interpretation:

• setuid + setgid are active (from the leading 6).
• Owner and group have rw-.
• Others have rwx.

So this is not a mistake — it’s simply the complete encoding, where the leading octal digit represents privilege bits.

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🧱 4. When to See 3 vs 4 Digits

Context	Digits Used	Why
Normal file (chmod 755, ls -l)	3	No special bits → leading 0 omitted
Special binary (e.g., passwd, sudo)	4	setuid or setgid active
Secure directories like /tmp	4	Sticky bit active (e.g. 1777)

So in the sheet:

• Rows with 000X → ordinary permissions (same as 3-digit form).
• Rows with 1XXX, 2XXX, 4XXX, etc. → special bits applied.

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✅ 5. TL;DR Summary

Octal Digit Position	Meaning	Bit Flags
1st (thousands place)	Special bits	4 = setuid, 2 = setgid, 1 = sticky
2nd	Owner perms	4 = read, 2 = write, 1 = exec
3rd	Group perms	same
4th	Others perms	same

Examples:

Octal	Symbolic	Meaning
0644	rw-r--r--	Normal text file
0755	rwxr-xr-x	Executable file
1777	rwxrwxrwt	Sticky shared directory (/tmp)
4755	rwsr-xr-x	setuid program (runs as owner)
2755	rwxr-sr-x	setgid program (runs as group)
6667	rwSrwSrwx	both setuid/setgid + full others access

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In short: 👉 3-digit = regular permissions 👉 4-digit = full mode (including privilege bits) 👉 The sheet correctly uses 4-digit form (SXYZ) so you can represent all 4096 possible combinations — not just the 512 basic ones.

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[!NOTE] ### NOTE:

This is just provided as conceptual research, documentation, for informational-purposes only, etc., and has not been fully battle tested or vetted, however would appreciate hearing and learning about any implementations, and shared learnings. (Unless otherwise explicitly stated by the author.)

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