User:Cloudy176/Department of bubbly negative numberbottles/Size N Arrays

My archive of Size N Arrays pages.

= Size 2 Arrays =

Rules
{a,1} = a

{a,b} = a*{a,b-1}

Examples
{a} = a

{a,b} = ab

{a,a} = aa = 2a

{5,5} = 3125

{10,10} = ten billion

{10,100} = googol

{100,100} = 10200

{10,googol} = googolplex

Continuing in this way, we can form a number of astronomical scale.

= Size 3 Arrays =

{a} = a

{a,b} = ab, That size 2.

Rules
{a,1,c} = a

{a,b,1} = ab

{a,b,c} = {a,{a,b-1,c},c-1}

Examples
{1,1,1} = 1

{2,2,2} = 2^^2 = 4

{3,3,3} = 3^^^3 = 3^^3^^3 = 3^^7625597484987 = 3^3^3^3^…^3^3^3 (7625597484987 3's) = tritri, this number has about 3^^7625597484986 digits.

{4,4,4} = 4^^^^4 = 4^^^4^^^4^^^4 = 4^^^4^^^4^^4^^4^^4 = 4^^^4^^^4^^4^^4^4^4^4 = 4^^^4^^^4^^(4^4^4^...^4^4^4 (with 4^4^4^4 4's))

= 4^4^4^4^...^4^4^4^4, with 4^4^4^4^...^4^4^4^4 4's, with 4^4^4^4^...^4^4^4^4 4's, with ............. with 4 4's There are A1 power towers. A1 has A2 power towers, A4 has 4 towers. This is tritet.

{5,5,5} = 5^^^^^5 = 5^^^^5^^^^5^^^^5^^^^5

= 5^5^5^...^5^5^5, with 5^5^5^...^5^5^5 5's, with 5^5^5^...^5^5^5 5's, with 5^5^5^...^5^5^5 5's, with........, with 5 5's There are A1 towers. A1 has A2 power towers, A2 has A3 power towers, A's in base 5. AAAAA has 5 towers. This is tripent.

{n,n,n} = n^^^^^^^...........n ^'s..........^^^^^^^n = {n,2,1,2}

{4,2,4} = 4^^^^2 = 4^^^4 = 4^^4^^4^^4 = 4^^4^^4^4^4^4 = 4^4^4^4^....^4^4^4^4 with 4^4^4^...^4^4^4 4's, with 4^4^4^4 4's

{a,a,b} = a ^^^^^^^...........b ^'s..........^^^^^^^ a

{a,b,c} = a ^^^^^^^...........c ^'s..........^^^^^^^ b

{6,6,6} = 6^^^^^^6 = trihex = AAAAAAA......AAAAAAA, with AAAAAAA......AAAAAAA A's, with AAAAAAA......AAAAAAA A's, with AAAAAAA......AAAAAAA A's, with AAAAAA A's. A's in base 6.

= Size 4 Arrays =

{a} = a

{a,b} = ab,That size 2.

{a,b,c} = a^^^^^^^^^^...c ^'s...^^^^^^^^^^b,That size 3.

Rules
{a,1,c,d} = a

{a,b,c,1} = {a,b,c}

{a,b,c,d} = {a,{a,b-1,c,d},c-1,d}

{a,b,1,d} = {a,a,{a,b-1,c,d},d-1}

General functions
Row 1: 3

Row 2: 3^^^3

Row 3: 3^^^^^````^^^^^3, 3^^^3 ^'s

Row 4: 3^^^^^````^^^^^3, 3^^^^^````^^^^^3 ^'s, 3^^^3 ^'s

etc.

Go to row 3^^^^^````^^^^^3 (3^^^3 ^'s), This is {3,3,2,2} = 3{3{3{3{...3{3}3...}3}3}3}3 (3{3{3}3}3 3's from center)

Go all the way to row {3,3,2,2}, this is only {3,3,3,2}

Let H = {3,1,3,2}, HH = {3,2,3,2}, HHH = {3,3,3,2}

L1: H

L2: HHH

L3: HHHHHHH...HHHHHHH (HHH H's)

L4: HHHHHHHHHHHHHHH...HHHHHHHHHHHHHHHH (HHHHHHH...HHHHHHH (HHH H's))

L5: HHHHHHHHHHHHHHHHHHHHHHH...HHHHHHHHHHHHHHHHHHHHHHHHHHH (HHHHHHHHHHHHHHH...HHHHHHHHHHHHHHHH (HHHHHHH...HHHHHHH (HHH H's)))

Go to LL, X is equal to HHH...HHH (HHH H's)

Go to LX, which is LHHHHHHHHHHHH...HHHHHHHHHHHHH (HHH H's), That is tetratri.

Examples
{a,b,1,2} = a^^^^^^^^^^...{a,b-1,1,2} ^'s...^^^^^^^^^^b

{1,2,3,4} = 1 { { { {3} } } } 2 = 1

{1,b,c,d} = 1

{2,b,c,d} = 2, if b = 1, 4 if b ≥ 2, actually we can prove.

{2,b,1,2} = {2,2,{2,b-1,1,2}} = 4, if there are 2 2s in the beginning its 4.

{2,b,2,2} = {2,{2,b-1,2,2},1,2} = 4 (previous example)

{2,b,3,2} = {2,{2,b-1,3,2},2,2} = 4 (previous example)

{2,b,3,infinity} = 4

{2,a,1,3} = {2,2,{2,a-1,1,3},2} = 4

{2,a,1,x} = {2,2,{2,a-1,1,x},x-1} = 4

{2,a,2,x} = {2,{2,a-1,2,x},1,x} = 4 (previous example)

The 3,4,5,... do the same thing as above so we proved it.

{2,a,infinity,infinity} = 4

{a,2,1,2} = a^^^^^^^^^^...a ^'s...^^^^^^^^^^a = {a,a,a}

{a,b,c,2} = G^^^^^^^^^^...c ^'s...^^^^^^^^^^b, G's in base a.

{a,b,c,3} = H^^^^^^^^^^...c ^'s...^^^^^^^^^^b, H's in base a.

{a,b,c,1} = {a,b,c}

{a,a,a,a} = {a,2,1,1,2}

{a,b,c,4} = I^^^^^^^^^^...c ^'s...^^^^^^^^^^b, I's in base a.

{3,2,1,2} = 3^^^3

{3,3,1,2} = 3^^^^^^^^^^...3^^^3 ^'s...^^^^^^^^^^3

{3,4,1,2} = 3^^^^^^^^^^...3^^^^^^^^^^ ...3^^^3 ^'s...^^^^^^^^^^3 ^'s... ^^^^^^^^^^3

{10,100,1,2} = 10^^^^^^^^^^...{10,99,1,2} ^'s...^^^^^^^^^^10 = corporal

{3,2,2,2} = {3,3,1,2}

{3,3,2,2} = {3,{3,2,2,2},1,2} = 3^^^^^^^^^^...3^^^^^^^^^^ ...3^^^^^^^^^^ .. .. .. .. ..3 ^'s.... .. .. .. ^^^^^^^^^^3 ^'s... ^^^^^^^^^^3 ^'s... ^^^^^^^^^^3, a tower of height {3,2,2,2}

{3,4,2,2} = 3^^^^^^^^^^...3^^^^^^^^^^ ...3^^^^^^^^^^ .. .. .. .. ..3 ^'s.... .. .. .. ^^^^^^^^^^3 ^'s... ^^^^^^^^^^3 ^'s... ^^^^^^^^^^3, a tower of height 3^^^^^^^^^^...3^^^^^^^^^^ ...3^^^^^^^^^^ .. .. .. .. ..3 ^'s.... .. .. .. ^^^^^^^^^^3 ^'s... ^^^^^^^^^^3 ^'s... ^^^^^^^^^^3, a tower of height {3,2,2,2}

{3,5,2,2} = a tower of height {3,4,2,2}

{3,b,2,2} = a tower of height {3,b-1,2,2}

{3,65,1,2} < Graham's Number

{3,66,1,2} > Graham's Number

= Size 5 Arrays =

{a} = a

{a,b} = ab, That size 2

{a,b,c} = a {c} b, That size 3

{a,b,c,d} = a {...{c}...} b, with d sets of {}, That size 4.

Rules
{a,b,c,d,1} = {a,b,c,d}

{a,1,c,d,e} = a

{a,b,c,d,e} = {a,{a,b-1,c,d,e},c-1,d,e}

{a,b,1,d,e} = {a,a,{a,b-1,1,d,e},d-1,e}

{a,b,1,1,e} = {a,a,a,{a,b-1,c,1,e},e-1}

Examples
{3,2,1,1,2} = tetratri

{3,3,1,1,2} = 3 3, with tetratri sets of {}

{3,4,1,1,2} = 3 3, with {3,3,1,1,2} sets of {}

{3,2,2,1,2} = 3 3, with tetratri sets of {}

{3,3,2,1,2} = 3 3, with 3 {{} 3 sets of {}, with 3 {{} 3 sets of {}, with 3 {{} 3 sets of {}, with 3 {{} 3 sets of {}, with 3 {{} 3 sets of {}, with 3 …………………… {}. This is repeated {3,2,2,1,2} times!!!!!!!!!!

{3,2,3,1,2} = 3 3, with 3 {{} 3 sets of {}, with 3 {{} 3 sets of {}, with 3 {{} 3 sets of {}, with 3 {{} 3 sets of {}, with 3 {{} 3 sets of {}, with 3 …………………… {}. This is repeated {3,2,2,1,2} times!!!!!!!!!!

{3,3,3,1,2} = {3,2,1,2,2} = 3 3, with 3 {{} 3 sets of {}, with 3 {{} 3 sets of {}, with 3 {{} 3 sets of {}, with 3 {{} 3 sets of {}, with 3 {{} 3 sets of {}, with 3 …………………… {}. This is repeated L1 times, where L1 is 3 3, with 3 {{} 3 sets of {}, with 3 {{} 3 sets of {}, with 3 {{} 3 sets of {}, with 3 {{} 3 sets of {}, with 3 {{} 3 sets of {}, with 3 …………………… {}. This is repeated L2 times, where L2 is 3 3, with 3 {{} 3 sets of {}, with 3 {{{{{…………}}}} 3 sets of {}, with 3 {{{{{……{}……}}}} 3 sets of {}, with 3 {{{{{…………}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 …………………… {}. This is repeated L3 times,..., where L({3,2,3,1,2}) is 3!!! YIKES!!!

{3,4,2,1,2} = 3 {{{{{……{{{{{3}}}}}……}}}}} 3, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 …………………… {}. This is repeated A times, where A is 3 {{{{{……{{{{{3}}}}}……}}}}} 3, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 …………………… {}. This is repeated {3,2,2,1,2} times.

{3,5,2,1,2} = 3 {{{{{……{{{{{3}}}}}……}}}}} 3, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 …………………… {}. This is repeated A times, where A is 3 {{{{{……{{{{{3}}}}}……}}}}} 3, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 …………………… {}. This is repeated B times, where B is 3 {{{{{……{{{{{3}}}}}……}}}}} 3, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 …………………… {}. This is repeated {3,2,2,1,2} times.

{3,6,2,1,2} = 3 {{{{{……{{{{{3}}}}}……}}}}} 3, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 …………………… {}. This is repeated A times, where A is 3 {{{{{……{{{{{3}}}}}……}}}}} 3, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 …………………… {}. This is repeated B times, where B is 3 {{{{{……{{{{{3}}}}}……}}}}} 3, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 …………………… {}. This is repeated C times, where C is 3 {{{{{……{{{{{3}}}}}……}}}}} 3, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 …………………… {}. This is repeated {3,2,2,1,2} times.

{3,7,2,1,2} = 3 {{{{{……{{{{{3}}}}}……}}}}} 3, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 …………………… {}. This is repeated A times, where A is 3 {{{{{……{{{{{3}}}}}……}}}}} 3, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 …………………… {}. This is repeated B times, where B is 3 {{{{{……{{{{{3}}}}}……}}}}} 3, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 …………………… {}. This is repeated C times, where C is 3 {{{{{……{{{{{3}}}}}……}}}}} 3, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 …………………… {}. This is repeated {3,3,2,1,2} times.

{3,8,2,1,2} = 3 {{{{{……{{{{{3}}}}}……}}}}} 3, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 …………………… {}. This is repeated A times, where A is 3 {{{{{……{{{{{3}}}}}……}}}}} 3, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 …………………… {}. This is repeated B times, where B is 3 {{{{{……{{{{{3}}}}}……}}}}} 3, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 …………………… {}. This is repeated C times, where C is 3 {{{{{……{{{{{3}}}}}……}}}}} 3, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 …………………… {}. This is repeated {3,4,2,1,2} times.

...

{3,n,2,1,2} = 3 {{{{{……{{{{{3}}}}}……}}}}} 3, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 …………………… {}. This is repeated {3,n-1,2,1,2} times!!!!!!!!!!

{3,n,3,1,2} = 3 {{{{{……{{{{{3}}}}}……}}}}} 3, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 …………………… {}. This is repeated L1 times, where L1 is 3 {{{{{……{{{{{3}}}}}……}}}}} 3, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 …………………… {}. This is repeated L2 times, where L2 is 3 {{{{{……{{{{{3}}}}}……}}}}} 3, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 {{{{{……{{{{{3}}}}}……}}}} 3 sets of {}, with 3 …………………… {}. This is repeated L3 times,..., where L({3,n-1,3,1,2}) is 3!!!

The first three entries are like Ackermann style recursion.

Now change d to 2.

{3,2,1,2,2} = {3,3,3,1,2} 3 levels of recursion past the {}.

{3,2,2,2,2} = {3,3,1,2,2} = {3,3,{3,2,1,2,2},1,2} this is {3,2,1,2,2} levels of recursion past {}.

{a,b,1,2,2} = {a,b-1,1,2,2} levels of recursion past {}.

{3,3,3,2,2} = {3,2,1,3,2}

{3,3,2,2,2} = {3,{3,3,1,2,2},1,2,2}

Now d = 3.

{3,3,3,3,2} = {3,2,1,1,3}

{3,1,1,3,2} = 3

{3,4,3,3,2} = {3,{3,2,1,1,3},2,3,2}

{10,10,10,100,2} = triggol

Beyond e = 2.

Let's have a function G where G = 3, GG = 3{{{3}}}3, this is tetratri, GGG = 3{{{{{{{{.......{{{{{{3}}}}}}.......}}}}}}}}3, with tetratri sets of {}, Go to GGGGGGGGGGGGGG..............GGG G's...............GGGGGGGGGGGGGG, this is {3,3,2,1,2}

GG G = {3,3,2,1,2}

GG G G G ... G G      (GG G G's) = {3,3,3,1,2} = G^^^3

G^^^^3 = {3,3,4,1,2}, G^^^^^3 = {3,3,5,1,2}, G^^^^^^3 = {3,3,6,1,2}, G^^^^^^^^......G^^^G ^'s......^^^^^G = {3,3,1,2,2}

GG = {3,3,3,2,2}

GG = {3,3,3,3,2}

GG with GG sets of {}, This is {3,3,{3,3,3,3,2},2,2} = {3,3,3,{3,2,1,1,3},2} = {3,3,1,1,3}, we are now breaking past the G function! We need a new function, call it H. H = G, HH = GG, HHH = GG with GG sets of {}, we can now describe {3,3,1,1,3} as HHH or H^3. You probably see the pattern.

H^^H = {3,3,2,1,3}

H^^^H = {3,3,3,1,3}

H^^^^^^......H^^^H ^'s......^^^^^^^H = {3,3,1,2,3}

H^^^^^^......H^^^^^^^ ......H^^^H ^'s......^^^^^^^H ^'s...... ^^^^^^^H = {3,4,1,2,3} This is {3,3,1,2,3} levels of recursion past the H function.

H{n}H = n levels of recursion past the H function, {3,3,n,1,3}

HH = {3,3,n,2,3}

HH = {3,3,n,3,3}, make n=3, this is pentatri

Rules
{a,1,c,d,e,f} = a

{a,b,c,d,e,1} = {a,b,c,d,e}

{a,b,c,d,e,f} = {a,{a,b-1,c,d,e,f},c-1,d,e,f}

{a,b,1,d,e,f} = {a,a,{a,b-1,c,d,e,f},d-1,e,f}

{a,b,1,1,e,f} = {a,a,a,{a,b-1,c,d,e,f},e-1,f}

{a,b,1,1,1,f} = {a,a,a,a,{a,b-1,c,d,e,f},f-1}

Examples
{a,b,c,1,e,f} = {a,b,c,{a,b-1,c,1,e},e-1,f}

{a,1,c,d,e,f} = a

{2,2,2,2,2,2} = 4

{3,2,4,2,1,5} = {3,{3,3,3,2,1,5},3,1,5}

{3,3,1,1,2,2} = {3,{3,2,1,1,1,2},1,1,2}

{3,3,3,3,3,3} = Hexatri

{3,2,1,1,1,2} = {3,3,3,3,3} = pentatri

{3,3,1,1,1,2} = {3,3,3,3,pentatri} = pentatriplex

{3,4,1,1,1,2} = {3,3,3,3,pentatriplex} = pentatriduplex

{3,2,2,1,1,2} = {3,3,1,1,1,2} = pentatriplex

{3,3,2,1,1,2} = {3,pentatriplex,1,1,1,2} = pentatridex??

{3,4,2,1,1,2} = {3,pentatridex??,1,1,1,2} = pentatridudex??

{3,2,3,1,1,2} = {3,3,2,1,1,2}

{3,3,3,1,1,2} = {3,pentatridex??,2,1,1,2} = pentatrithrex????

{3,2,1,2,1,2} = {3,3,3,1,1,2} = pentatrithrex???

{3,3,1,2,1,2} = {3,3,pentatrithrex???,1,1,2} = pentatri-(pentatrithrex)-ex???? or pentatri-suplex?????

= Size 7 Arrays =

{3,3,3,3,3,3,3} = heptatri

{4,4,4,4,4,4,4} = heptatet

{2,3,1,1,1,1,2} = 4?

{6,4,2,5,3,7,7} > heptahex

The smallest non-trivial

{3,2,1,1,1,1,2} = {3,3,3,3,3,{3,1,1,1,1,1,2},1} = {3,3,3,3,3,3} = hexatri

{3,3,1,1,1,1,2} = {3,3,3,3,3,{3,2,1,1,1,1,2},1}