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A072233
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Square array T(n,k) read by antidiagonals giving number of ways to distribute n indistinguishable objects in k indistinguishable containers; containers may be left empty.
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88
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1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1, 2, 2, 1, 1, 0, 1, 3, 3, 2, 1, 1, 0, 1, 3, 4, 3, 2, 1, 1, 0, 1, 4, 5, 5, 3, 2, 1, 1, 0, 1, 4, 7, 6, 5, 3, 2, 1, 1, 0, 1, 5, 8, 9, 7, 5, 3, 2, 1, 1, 0, 1, 5, 10, 11, 10, 7, 5, 3, 2, 1, 1, 0, 1, 6, 12, 15, 13, 11, 7, 5, 3, 2, 1, 1, 0, 1, 6, 14, 18, 18, 14, 11, 7, 5, 3, 2, 1, 1
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OFFSET
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0,13
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COMMENTS
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T(n,k) is also the number of partitions of n with greatest part k, if we assume the greatest part of an empty partition to be 0. Row n = 9 counts the following partitions:
111111111 22221 333 432 54 63 72 81 9
222111 3222 441 522 621 711
2211111 3321 4221 531 6111
21111111 32211 4311 5211
33111 42111 51111
321111 411111
3111111
(End)
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LINKS
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FORMULA
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T(0, k) = 1, T(n, 0) = 0 (n>0), T(1, k) = 1 (k>0), T(n, 1) = 1 (n>0), T(n, k) = 0 for n < 0, T(n, k) = Sum[ T(n-k+i, k-i), i=0...k-1] Or, T(n, 1) = T(n, n) = 1, T(n, k) = 0 (k>n), T(n, k) = T(n-1, k-1) + T(n-k, k).
G.f. Product_{j=0..infinity} 1/(1-xy^j). Regarded as a triangular array, g.f. Product_{j=1..infinity} 1/(1-xy^j). - Franklin T. Adams-Watters, Dec 18 2006
O.g.f. of column No. k of the triangle a(n,k) is x^k/product(1-x^j,j=1..k), k>=0 (the undefined product for k=0 is put to 1). - Wolfdieter Lang, Dec 03 2012
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EXAMPLE
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Table begins (upper left corner = T(0,0)):
1 1 1 1 1 1 1 1 1 ...
0 1 1 1 1 1 1 1 1 ...
0 1 2 2 2 2 2 2 2 ...
0 1 2 3 3 3 3 3 3 ...
0 1 3 4 5 5 5 5 5 ...
0 1 3 5 6 7 7 7 7 ...
0 1 4 7 9 10 11 11 11 ...
0 1 4 8 11 13 14 15 15 ...
0 1 5 10 15 18 20 21 22 ...
There is 1 way to distribute 0 objects into k containers: T(0, k) = 1. The different ways for n=4, k=3 are: (oooo)()(), (ooo)(o)(), (oo)(oo)(), (oo)(o)(o), so T(4, 3) = 4.
The triangle a(n,k) = T(n-k,k) begins:
n\k 0 1 2 3 4 5 6 7 8 9 10 ...
00 1
01 0 1
02 0 1 1
03 0 1 1 1
04 0 1 2 1 1
05 0 1 2 2 1 1
06 0 1 3 3 2 1 1
07 0 1 3 4 3 2 1 1
08 0 1 4 5 5 3 2 1 1
09 0 1 4 7 6 5 3 2 1 1
10 0 1 5 8 9 7 5 3 2 1 1
...
Row n=5 is, for k=1..5, [1,2,2,1,1] which gives the number of partitions of n=5 with k parts. See A008284 and the Franklin T. Adams-Watters comment above. (End)
Row n = 9 counts the following partitions:
9 54 333 3222 22221 222111 2211111 21111111 111111111
63 432 3321 32211 321111 3111111
72 441 4221 33111 411111
81 522 4311 42111
531 5211 51111
621 6111
711
(End)
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MATHEMATICA
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Flatten[Table[Length[IntegerPartitions[n, {k}]], {n, 0, 20}, {k, 0, n}]] (* Emanuele Munarini, Feb 24 2014 *)
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PROG
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(Sage)
from sage.combinat.partition import number_of_partitions_length
[[number_of_partitions_length(n, k) for k in (0..n)] for n in (0..10)] # Peter Luschny, Aug 01 2015
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CROSSREFS
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Sum of antidiagonal entries T(n, k) with n+k=m equals A000041(m).
The version for factorizations is A316439.
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KEYWORD
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AUTHOR
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Martin Wohlgemuth (mail(AT)matroid.com), Jul 05 2002
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EXTENSIONS
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STATUS
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approved
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