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A225622
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A(n,k) is the total number of parts in the set of partitions of an n X k rectangle into integer-sided squares, considering only the list of parts; square array A(n,k), n>=1, k>=1, read by antidiagonals.
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4
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1, 2, 2, 3, 5, 3, 4, 9, 9, 4, 5, 15, 16, 15, 5, 6, 21, 31, 31, 21, 6, 7, 30, 47, 59, 47, 30, 7, 8, 38, 73, 102, 102, 73, 38, 8, 9, 50, 101, 170, 156, 170, 101, 50, 9, 10, 60, 142, 250, 307, 307, 250, 142, 60, 10, 11, 75, 185, 375, 460, 529, 460, 375, 185, 75, 11
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OFFSET
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1,2
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LINKS
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FORMULA
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A(n,2) = A195014(n) = (n+1)(5n+3)/8 when n is odd
and 5n(n+2)/8 when n is even.
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EXAMPLE
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The square array starts:
1 2 3 4 5 6 7 8 9 10 11 12 ...
2 5 9 15 21 30 38 50 60 75 87 105 ...
3 9 16 31 47 73 101 142 185 244 305 386 ...
4 15 31 59 102 170 250 375 523 726 962 ...
5 21 47 102 156 307 460 711 1040 1517 ...
6 30 73 170 307 529 907 1474 2204 ...
7 38 101 250 460 907 1351 2484 ...
8 50 142 375 711 1474 2484 ...
9 60 185 523 1040 2204 ...
...
A(3,2) = 9 because there are 9 parts overall in the 2 partitions of a 3 X 2 rectangle into squares with integer sides. One partition comprises 6 1 X 1 squares and the other 2 1 X 1 squares and 1 2 X 2 square giving 9 parts in total.
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MAPLE
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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]$j=i+1..nops(l)]))
od; s
fi
end:
A:= (n, k)-> add(coeff(add(j, j=b(max(n, k),
[0$min(n, k)])), x, i), i=1..n):
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MATHEMATICA
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$RecursionLimit = 1000; 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, True, k++, If[l[[k]] == 0, Break[]]]; s = {}; For[i = k, i <= Length[l] && l[[i]] == 0, i++, s = s ~Union~ Map[Function[{v}, v + x^(1+i-k) ], b[n, Join[l[[1 ;; k-1]], Array[1+i-k&, i-k+1], l[[i+1 ;; -1]]]]]]; s]]; A[n_, k_] := Sum[Coefficient[Sum[j, {j, b[Max[n, k], Array[0&, Min[n, k]]]}], x, i], {i, 1, n}]; Table[Table[A[n, 1+d-n], {n, 1, d}], {d, 1, 15}] // Flatten (* Jean-François Alcover, Mar 06 2015, after Alois P. Heinz *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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