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

lori_m 

I wish I knew for how long these workarounds are to be needed.  I suspect the vast majority of spreadsheet users will never make the jump to the Excel formula as a functional programming language and the nested array problem only makes it more intimidating!

Towers of Hanoi.  I am not sure which discussion it was attached to but it was posted by Patrick (I have a copy).  I do not know why it has disappeared, though.  The algorithm appears to build a table of moves that lead to the solution but without ever calculating the current state of the problem.

BTW I am planning to attend EuSpRIG Annual Conference | European Spreadsheet Risk Interest Group. 

The programme looks quite varied this year.

Ok, so after several weeks of head banging, i feel like there may finally be some progress!

First, as far as i'm aware the list of scalable alternatives to the basic REDUCE/VSTACK method for stacking arrays now includes: nested reduce, recursive bisection, thunk arrays and lambda accumulators. Several examples are given in earlier comments; a relatively efficient implementation utilising lambda accumulators could be

BYROWλ
=LAMBDA(array, function,
LET(
    n, ROWS(array),
    k, CEILING.MATH(LOG(n,2)),
    REDUCE(
        LAMBDA(r,function(CHOOSEROWS(array,r))),
        SEQUENCE(k,,0),
        LAMBDA(a,i,LAMBDA(j,IF(j+2^i<=n,VSTACK(a(j),a(j+2^i)),a(j))))
        )(1)
    )
)

Second, for enumerating combinations, tests suggest lookups can be faster than stacking when returning longer result sets and that 2D-arrays may be needed to overcome decimal precision limitations such as:

=LAMBDA(n,k,
REDUCE(SEQUENCE(n-k+1),SEQUENCE(k-1,,2),LAMBDA(a,j,
    LET(r,ROUND(COMBIN(SEQUENCE(n-k+1,,j),j),),
        i,SEQUENCE(@TAKE(r,-1,1)),
        HSTACK(
            CHOOSEROWS(a,i-XLOOKUP(i,r,VSTACK(0,DROP(r,-1)),,1)),
            XMATCH(i,r,1)+j-1
        )
    )
)))(20,10)

P.S. was there a Towers of Hanoi lambda posted here previously? Maybe I was hallucinating but I had been hoping to check it out when it seemed to vanish!

PeterBartholomew1 

That thunk method looks pretty neat - I was impressed by the speed too. In previous attempts working with thunks I had experienced performance issues that were difficult to circumvent. Passing the thunk array as an array parameter to SCAN is clever and appears much more efficient than passing as a lambda parameter - within MAKEARRAY for example.

My first attempt to improve efficiency was to apply memoization which is a standard technique to avoid recomputing values within the recursion. I started looking at converting some online code examples to lambdas by building a table of thunks before realising that this was going to be similar to your method but probably less fast!

A follow-up thought was to generate all decimal combinations in a single row of Pascal's triangle iteratively,

=SORT(REDUCE(0,SEQUENCE(n),LAMBDA(a,_,VSTACK(a*{1,2},a*{1,2}+1))),{1,2})

With n = 12 for example this returns 2^12=4096 rows, the first column is the number of combinations k and the second is the decimal representation of the combination. To return just the COMBIN(n,k) combinations we only need keep combinations less than k,

=LAMBDA(n, k,
    LET(
        combs, REDUCE(
            0,
            SEQUENCE(n),
            LAMBDA(a, i,
                SORT(
                    VSTACK(
                        DROP(a, IF(i > n - k, INT(COMBIN(i - 1, n - k)), 0)) * {1, 2},
                        DROP(a, IF(i > k, -INT(COMBIN(i - 1, k)), 0)) * {1, 2} + 1
                    ),
                    {1, 2}
                )
            )
        ),
        BASE(TAKE(combs, , -1), 2, n)
    )
)

I expected this might be faster than the thunk approach but in practice it fell short due to the repeated SORT operations so likely needs further refinements. I was also surprised that COMBIN returns non-integers in some cases (e.g. try removing 'INT' with n=12 and k=6)