How to create grouped serial numbers using Arrayformula?

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What would be the Array formulas at A1, B1, C1 & D1 (separately) to give the desired output columns as shown.

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NOTE: Difference between "Main Headings" (RELATIVES, FRIENDS, BUSINESS FRIENDS) & "names" is that the "Main Headings" has 1 space letter at the start & end.

Column A,B,C all three are same the only difference is that for the "Main Headings" Column A gives Alphabets, Column B gives Numbers & Column C gives Roman Numerals.

CONDITION FOR SERIAL NUMBERING:
(1) Serial numbering is done completely based on Columns: NAMES & PAX.
(2) If PAX is 0 then output is blank.

FOR Column A, B, C:
When PAX is blank:
-> if it's a "Main heading" then output is a Alphabet OR
-> if it's a "name" then the output is blank.

FOR Column D:
When PAX is blank:
-> if it's a "Main heading" then output is a blank OR
-> if it's a "name" then the output is serial Numbering until the next "Main Heading", then again restarts from 1 until next "Main Heading" & so on.

EDIT:
Column NAMES & PAX are input, rest are the required output.

Data:

This formula in A2 generates the result for all the columns.

=ARRAYFORMULA(LET(
   names, E2:E, pax, F2:F, i, ROW(names),
   header_count, COUNTIFS(RIGHT(names), " ", i, "<=" & i),
   serial_numbering, COUNTIFS(names, "<>", pax, "<>0", i, "<=" & i),
   result, IFNA(
     IF(RIGHT(names)=" ", 
       {
        CHAR(header_count + 64),
        header_count,
        ROMAN(header_count)
       },
       IF(pax = 0, ,
         {
          serial_numbering + {0, 0, 0},
          serial_numbering - XLOOKUP(header_count, header_count, serial_numbering)
         }
       )
     )
   ), 
   result
 ))

How it works

First, we do a running count of the headers and store it in header_count:

header_count ← COUNTIFS(RIGHT(names), " ", i, "<=" & i),

This formula returns an array with ROWS(names) rows where each row is the count of the headers encountered so far. A header is any value in the 'names' column that ends with a space " ".

Then, we do the numbering based on the conditions mentioned in the OP (namely, name not empty and pax not 0) and assign it to serial_numbering:

serial_numbering ← COUNTIFS(names, "<>", pax, "<>0", i, "<=" & i)

Then, we go through each value in 'names'.

If it's a header, we return the headers (Alphabet, Number, Roman) based on the current header_count:

IF(RIGHT(names)=" ", 
   {
    CHAR(header_count + 64),
    header_count,
    ROMAN(header_count)
   },

If it's not a header, and if pax is not 0 (or blank*), we return:

{
 serial_numbering + {0, 0, 0},
 serial_numbering - XLOOKUP(header_count,   header_count, serial_numbering)
}

where

serial_numbering + {0, 0, 0}

is shorthand for

{serial_numbering, serial_numbering, serial_numbering}

and

serial_numbering - XLOOKUP(header_count, header_count, serial_numbering)

is the current serial number minus the serial number associated to the current header. The subtraction is necessary to reset the serial number everytime we encounter a new header.

^{* Because of type coercion value = 0 is equivalent to OR(value = 0, ISBLANK(value))}