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A221640 Number T(n,k) of different numbers of square parts in the set of partitions of an n X k rectangle into squares with integer sides, considering only the list of parts; triangle T(n,k), 1<=k<=n, read by rows. 1
1, 1, 2, 1, 2, 3, 1, 3, 4, 7, 1, 3, 5, 9, 11, 1, 4, 7, 12, 18, 23, 1, 4, 8, 15, 23, 30, 34, 1, 5, 10, 20, 27, 37, 43, 52, 1, 5, 12, 22, 32, 42, 50, 58, 68, 1, 6, 14, 27, 36, 47, 57, 68, 76, 87, 1, 6, 16, 30, 42, 54, 64, 75, 85, 96, 105 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

The triangle begins:

. k  1    2    3    4    5    6    7    8

n

1    1

2    1    2

3    1    2    3

4    1    3    4    7

5    1    3    5    9   11

6    1    4    7   12   18   23

7    1    4    8   15   23   30   34

8    1    5   10   20   27   37   43   52

LINKS

Alois P. Heinz, Rows n = 1..14, flattened

Christopher Hunt Gribble, C++ program

EXAMPLE

T(4,3) = 4 because there are 4 partitions of a 4 X 3 rectangle into integer sided squares with different numbers of parts:

Partition                           Number of parts

12 1 X 1 squares                           12

.8 1 X 1 squares, 1 2 X 2 square            9

.4 1 X 1 squares, 2 2 X 2 squares           6

.3 1 X 1 squares, 1 3 X 3 square            4

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+1, 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, k)-> nops(b(max(n, k), [0$min(n, k)])):

seq(seq(T(n, k), k=1..n), n=1..10);  # Alois P. Heinz, Aug 08 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[#+1&, b[n, Join[ l[[1 ;; k-1]], Array[1+i-k&, i-k+1], l[[i+1 ;; Length[l]]]]]]]; s]]; T[n_, k_] := Length[b[Max[n, k], Array[0&, Min[n, k]]]]; Table[Table[ T[n, k], {k, 1, n}], {n, 1, 10}] // Flatten (* Jean-Fran├žois Alcover, Jan 24 2016, after Alois P. Heinz *)

CROSSREFS

Diagonal = A226937.

Cf. A224697, A227998.

Sequence in context: A330661 A091438 A011794 * A073300 A104468 A293003

Adjacent sequences:  A221637 A221638 A221639 * A221641 A221642 A221643

KEYWORD

nonn,tabl

AUTHOR

Christopher Hunt Gribble, Aug 08 2013

EXTENSIONS

More terms from Alois P. Heinz, Aug 08 2013

STATUS

approved

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Last modified May 10 14:22 EDT 2021. Contains 343770 sequences. (Running on oeis4.)