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A226912 Irregular triangle T(n,k) is the frequency with which the number of square parts equals k in each partition of an n X n square lattice into squares, considering only the list of parts. 3
1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 3, 0, 1, 4, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 2, 0, 2, 2, 2 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,64

COMMENTS

The sequence was derived from the documents in the Links section.  The documents are first specified in the Links section of A034295.

The irregular triangle is presented below.

  k  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20 ...

n

1    1

2    1  0  0  1

3    1  0  0  0  0  1  0  0  1

4    1  0  0  1  0  0  1  1  0  1  0  0  1  0  0  1

5    1  0  0  0  0  0  0  1  0  1  1  0  1  1  0  1  1  0  1  0 ...

6    1  0  0  1  0  1  0  0  3  0  1  4  1  1  2  1  1  2  1  1 ...

7    1  0  0  0  0  0  0  0  1  2  0  2  2  2  2  2  3  2  3  2 ...

8    1  0  0  1  0  0  1  1  0  3  3  2  5  3  3  8  5  4  7  6 ...

LINKS

Alois P. Heinz, Rows n = 1..13, flattened

Jon E. Schoenfield, Table of solutions for n <= 12

Alois P. Heinz, More ways to divide an 11 X 11 square into sub-squares

Alois P. Heinz, List of different ways to divide a 13 X 13 square into sub-squares

FORMULA

It appears that for n > 6, T(n, floor(n^2/2) + 3 : n^2) =  T(n-1, floor(n^2/2) - 2n + 4 : (n-1)^2).

EXAMPLE

For n = 3, the partitions are:

Square side 1 2 3 Number of parts

            9 0 0       9

            5 1 0       6

            0 0 1       1

So T(3,1) = 1, T(3,6) = 1, T(3,9) = 1.

MAPLE

b:= proc(n, l) option remember; local i, k, s, t;

      if max(l[])>n then {} elif n=0 or l=[] then {0}

    elif min(l[])>0 then t:=min(l[]); b(n-t, map(h->h-t, l))

    else for k do if l[k]=0 then break fi od; s:={};

         for i from k to nops(l) while l[i]=0 do s:=s union

             map(v->v+x^(1+i-k), b(n, [l[j]$j=1..k-1,

                 1+i-k$j=k..i, l[j]$j=i+1..nops(l)]))

         od; s

      fi

    end:

T:= n-> (p-> seq(coeff(p, z, i), i=1..n^2))

        (add(z^subs(x=1, f), f=b(n, [0$n]))):

seq(T(n), n=1..9);  # Alois P. Heinz, Jun 22 2013

MATHEMATICA

b[n_, l_List] := b[n, l] = Module[{i, k , s, t}, Which[Max[l]>n, {}, n==0 || l=={}, {0}, Min[l]>0, t = Min[l]; b[n-t, l-t], True, For[k=1, k <= Length[l], k++,  If[l[[k]]==0, Break[]]]; s={}; For[i=k, i <= Length[l] && l[[i]] == 0, i++, s = s ~Union~ Map[# + x^(1+i-k)&, b[n, Join[l[[1 ;; k-1]], Array[1+i-k&, i-k+1], l[[i+1 ;; Length[l]]]]]]]; s]]; T[n_] := Function[p, Table[Coefficient[p, z, i], {i, 1, n^2}]][Sum[z^(f /. x -> 1), {f, b[n, Array[0&, n]]}]]; Table[T[n], {n, 1, 9}] // Flatten (* Jean-Fran├žois Alcover, Jan 24 2016, after Alois P. Heinz *)

CROSSREFS

Row sums = A034295.

Row lengths are A000290.

Sequence in context: A178116 A238709 A245120 * A177330 A197126 A256987

Adjacent sequences:  A226909 A226910 A226911 * A226913 A226914 A226915

KEYWORD

nonn,tabf,hard

AUTHOR

Christopher Hunt Gribble, Jun 22 2013

STATUS

approved

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Last modified July 8 05:33 EDT 2020. Contains 335513 sequences. (Running on oeis4.)