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A189074
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Irregular triangle read by rows: T(n,k) = number of compositions of n with k inversions (n >= 0, 0 <= k <= floor(n^2/8).
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2
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1, 1, 2, 3, 1, 5, 2, 1, 7, 5, 3, 1, 11, 8, 7, 4, 2, 15, 15, 14, 10, 6, 3, 1, 22, 23, 26, 21, 17, 10, 6, 2, 1, 30, 37, 44, 42, 36, 27, 19, 11, 6, 3, 1, 42, 55, 73, 74, 73, 60, 50, 34, 24, 13, 8, 4, 2, 56, 83, 115, 128, 133, 123, 109, 87, 68, 48, 32, 20, 12, 6, 3, 1, 77, 118, 177, 209, 235, 230, 223, 192, 166, 129, 100, 70, 51, 31, 20, 11, 6, 2, 1
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OFFSET
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0,3
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COMMENTS
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Row sums are powers of 2.
The Heubach et. al. reference has a table for n <= 12.
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LINKS
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Alois P. Heinz, Rows n = 0..25, flattened
S. Heubach, A. Knopfmacher, M. E. Mays and A. Munagi, Inversions in Compositions of Integers, to appear in Quaestiones Mathematicae.
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EXAMPLE
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T(4,0) = 5: [4], [1,3], [2,2], [1,1,2], [1,1,1,1] - all partitions of 4.
T(5,2) = 3: [2,2,1], [3,1,1], [1,2,1,1].
T(6,4) = 2: [2,2,1,1], [2,1,1,1,1].
Triangle begins:
1
1
2
3 1
5 2 1
7 5 3 1
11 8 7 4 2
15 15 14 10 6 3 1
22 23 26 21 17 10 6 2 1
...
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MAPLE
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T:= proc(n) option remember; local b, p;
b:=proc(m, i, l)
if m=0 then p(i):= p(i)+1
else seq (b(m-h, i+nops (select (j->j<h, l)), [h, l[]]), h=1..m)
fi
end;
p:= proc() 0 end; forget(p);
b(n, 0, []); seq (p(i), i=0..floor(n^2/8))
end:
seq (T(n), n=0..12); # Alois P. Heinz, Apr 17 2011
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CROSSREFS
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The first two columns are A000041 and A058884 (for n>0). Lengths of rows are given by 1+A001972(n-3). Row sums are A011782.
Sequence in context: A214055 A066909 A095195 * A169615 A076791 A075014
Adjacent sequences: A189071 A189072 A189073 * A189075 A189076 A189077
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KEYWORD
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nonn,tabf
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AUTHOR
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N. J. A. Sloane, Apr 16 2011
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STATUS
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approved
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