Working with Binary numbers BASE(..., 2, 7) and bit operations BITXOR, BITAND

PeterBartholomew1

no words, what i have been through, despite thankful 🙂

=LET(
weight;{75\74\43\19\12\5\4};
binary;DEC2BIN(DROP(SEQUENCE(POWER(2;7));-1);7);
factor;MID(binary;SEQUENCE(1;7);1)*1;
votes;BYROW(weight*factor;LAMBDA(r;SUM(r)));
votemin;BYROW(weight*factor;LAMBDA(r;MIN(SUBSTITUTE(r;"0";"99")*1)));
detailed;HSTACK(binary;votes;votemin;(votes>121)*((votes-votemin)<122));
summary;FILTER(detailed;CHOOSECOLS(detailed;4));
header;{"involved"\"votesum"\"kingmaker"\"coalitions"};
divider;{"¤"\"¤"\"¤"\"¤"};
VSTACK(header;SORT(summary;2;1);divider;detailed))

PeterBartholomew1 

I've used BITXOR and BITAND very sparingly and I've played with BITLSHIFT and BITRSHIFT.  I understand what the functions do but have not used them enough to the point where I can conceptualize a solution for combinations/permutations.  I'll study your solution provided to see where I can find ways to make use of these functions.

My current way of tackling this task is with the help of DEC2BIN.  I understand the limitations involved but for 127 possibilities it's more than up to the task (for much more than 127 perhaps Python is the best option.)

'GenerateCombinations
=LAMBDA(party,seats,combinations,LET(
    header, HSTACK(TOROW(party), "Seats"),
    n, COUNT(seats),
    seq, SEQUENCE(n),
    combinations, SUM(COMBIN(n, seq)),
    BIN_vector, DEC2BIN(SEQUENCE(combinations), n),
    height, ROWS(BIN_vector),
    GetSeats, LAMBDA(r, c,
        LET(
            bin_val, MID(INDEX(BIN_vector, r), c, 1),
            seat_total, INDEX(seats, c),
            bin_val * seat_total
        )
    ),
    SeatMatrix, MAKEARRAY(height, n, GetSeats),
    CoalitionSeats, BYROW(SeatMatrix, SUM),
    CoalitionMatrix, FILTER(HSTACK(SeatMatrix, CoalitionSeats), CoalitionSeats >= required),
    Results, SORT(CoalitionMatrix, n + 1),
    VSTACK(header, Results)
))

Confession.

I have changed the code that checks whether a given proposed coalition contains a smaller viable coalition.  Originally I worked with arrays of the form {1,1,0,0,1,0,1} to check the subset condition and that required nested MAPs which were not pretty.  Since BITXOR and BITAND work with the decimal representation (i.e. scalars rather than arrays) that part could be simplified.

HasSubsetλ
"Tests whether each of an array of sets is a superset of the others"
= LET(
    q, TRANSPOSE(p),
    s, SIGN(IF(p <> q, BITAND(BITXOR(p, q), q) = 0)),
    BYROW(s, LAMBDA(x, SUM(x))) > 0
  )

Sadly, settling for the first thing that works has its shortcomings.

PeterBartholomew1 

Combinations is too complex for me, make the bookmark to play one day.

As for the binary logic long ago I used it to check if the character is in English alphabet or in digits

ABC = LAMBDA(chr,
    IFERROR(
        BITOR(chr, 48),
        IFERROR(CHAR(BITXOR(CODE(chr), 32)) = chr, 0)
    )
)

and only to play with it, sure could be done other way. No more practical cases in my practice.