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A230505 T(n,k,s) is the number of parts of each size in the set of partitions of an n X k rectangle into integer-sided squares with side s, considering only the list of parts; irregular triangle T(n,k,s), n >= k >= s >= 1, read by rows. 1
1, 2, 4, 1, 3, 8, 1, 14, 1, 1, 4, 12, 3, 27, 3, 1, 47, 10, 1, 1, 5, 18, 3, 41, 4, 2, 85, 13, 3, 1, 134, 16, 4, 1, 1, 6, 24, 6, 62, 7, 4, 135, 27, 5, 3, 250, 40, 13, 3, 1, 415, 82, 24, 6, 1, 1, 7, 32, 6, 87, 9, 5, 204, 34, 8, 4, 381, 53, 18, 5, 3, 717, 127, 45, 13, 4, 1 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,2
LINKS
Alois P. Heinz, Rows n = 1..78, flattened (Rows 1..42 from Christopher Hunt Gribble)
Christopher Hunt Gribble, C++ program
FORMULA
Sum_{s=1..k} T(n,k,s) = A225622(n,k).
Sum_{s=1..k} T(n,k,s)*s^2 = n*k*A224697(n,k).
EXAMPLE
T(5,4,2) = 13 because there are 13 2 X 2 squares in the 9 partitions of a 5 X 4 rectangle into integer-sided squares. The partitions are:
. Square side
. 1 2 3 4
1 20 0 0 0
2 16 1 0 0
3 12 2 0 0
4 8 3 0 0
5 4 4 0 0
6 11 0 1 0
7 7 1 1 0
8 3 2 1 0
9 4 0 0 1
Total 85 13 3 1
The irregular triangle begins:
n,k Square Side (s)
. 1 2 3 4 5 6 7 ...
1,1 1
2,1 2
2,2 4 1
3,1 3
3,2 8 1
3,3 14 1 1
4,1 4
4,2 12 3
4,3 27 3 1
4,4 47 10 1 1
5,1 5
5,2 18 3
5,3 41 4 2
5,4 85 13 3 1
5,5 134 16 4 1 1
6,1 6
6,2 24 6
6,3 62 7 4
6,4 135 27 5 3
6,5 250 40 13 3 1
6,6 415 82 24 6 1 1
7,1 7
7,2 32 6
7,3 87 9 5
7,4 204 34 8 4
7,5 381 53 18 5 3
7,6 717 127 45 13 4 1
7,7 1102 165 60 16 6 1 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]$j=i+1..nops(l)]))
od; s
fi
end:
T:= (n, k)->(p->seq(coeff(p, x, v), v=1..k))(add(h, h=b(n, [0$k]))):
seq(seq(T(n, k), k=1..n), n=1..9); # Alois P. Heinz, Oct 24 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_, k_] := Function[p, Table[Coefficient[p, x, v], {v, 1, k}]][b[n, Array[0&, k]] // Total]; Table[Table[T[n, k], {k, 1, n}], {n, 1, 9}] // Flatten (* Jean-François Alcover, Jan 24 2016, after Alois P. Heinz *)
CROSSREFS
Sequence in context: A248212 A290824 A272977 * A366062 A208526 A275895
KEYWORD
nonn,tabf
AUTHOR
STATUS
approved

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Last modified March 28 07:20 EDT 2024. Contains 371235 sequences. (Running on oeis4.)