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A228820
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Sum of positive F-ranks of all compositions of n. Also, sum of positive L-ranks of all compositions of n (see comments lines for definition).
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2
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0, 0, 1, 3, 9, 24, 60, 145, 342, 791, 1800, 4041, 8971, 19733, 43077, 93441, 201592, 432867, 925574, 1971633, 4185537, 8857634, 18691421, 39339638, 82599634, 173050951, 361825484, 755140789, 1573359111, 3273103135, 6799507189, 14106802811, 29231731788
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OFFSET
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0,4
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COMMENTS
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Here, the F-rank of a composition is defined by 2^(F-1) - N, where F is the first part and N is the number of parts. For example: the F-rank of the composition [6, 2, 1, 1] is (2^5 - 4) = 28.
Also, the L-rank of a composition is defined by 2^(L-1) - N, where L is the last part and N is the number of parts. For example: the L-rank of the composition [6, 2, 1, 1] is (2^0 - 4) = -3.
The sum of all F-ranks of all compositions of n is 0.
The sum of all L-ranks of all compositions of n is 0.
a(n) is also the sum of nonnegative terms in the n-th row of triangle A228821.
Note that in the table 1 (see example) the L-rank of the j-th composition is also the number of parts of the j-th region of the diagram minus the number of parts of the j-th composition. Also, note that in the table 2 the F-rank of the j-th composition is also the number of parts of the j-th region of the diagram minus the number of parts of the j-th composition. The same for all positive integers.
Also, the little F-rank of an overcomposition is defined by (2^(F-1) - N)/(2^D), where F is the first part, N is the number of parts and D is the number of distinct parts. For example: the little F-rank of the overcomposition [6, 2, 1, 1] is (2^5 - 4)/(2^3) = 7/2.
Also, the little L-rank of an overcomposition is defined by (2^(L-1) - N)/(2^D), where L is the last part, N is the number of parts and D is the number of distinct parts. For example: the little L-rank of the overcomposition [6, 2, 1, 1] is (2^0 - 4)/(2^3) = -3/8.
The sum of all little F-ranks of all overcompositions of n is 0.
The sum of all little L-ranks of all overcompositions of n is 0.
a(n) is also the sum of positive little F-ranks of all overcompositions of n.
a(n) is also the sum of positive little L-ranks of all overcompositions of n.
For the definition of overcomposition see A236002.
(End)
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LINKS
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EXAMPLE
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Table 1. Compositions of 4 in lexicographic order.
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j Composition Diagram F-rank L-rank
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. _ _ _ _
1 [1,1,1,1] | | | |_| 1 - 4 = -3 1 - 4 = -3
2 [1,1,2] | | |_ _| 1 - 3 = -2 2 - 3 = -1
3 [1,2,1] | | |_| 1 - 3 = -2 1 - 3 = -2
4 [1,3] | |_ _ _| 1 - 2 = -1 4 - 2 = 2
5 [2,1,1] | | |_| 2 - 3 = -1 1 - 3 = -2
6 [2,2] | |_ _| 2 - 2 = 0 2 - 2 = 0
7 [3,1] | |_| 4 - 2 = 2 1 - 2 = -1
8 [4] |_ _ _ _| 8 - 1 = 7 8 - 1 = 7
--- ---
Total sum: 0 0
Sum of positive terms: 9 9
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Table 2. Compositions of 4 in colexicographic order.
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j Composition Diagram F-rank L-rank
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. _ _ _ _
1 [1,1,1,1] |_| | | | 1 - 4 = -3 1 - 4 = -3
2 [2,1,1] |_ _| | | 2 - 3 = -1 1 - 3 = -2
3 [1,2,1] |_| | | 1 - 3 = -2 1 - 3 = -2
4 [3,1] |_ _ _| | 4 - 2 = 2 1 - 2 = -1
5 [1,1,2] |_| | | 1 - 3 = -2 2 - 3 = -1
6 [2,2] |_ _| | 2 - 2 = 0 2 - 2 = 0
7 [1,3] |_| | 1 - 2 = -1 4 - 2 = 2
8 [4] |_ _ _ _| 8 - 1 = 7 8 - 1 = 7
--- ---
Total sum: 0 0
Sum of positive terms: 9 9
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The sum of positive F-ranks of all compositions of 4 is 2+7 = 9, the same as the sum of positive L-ranks, so a(4) = 9.
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MAPLE
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a:= n-> add(add(binomial(n-k-1, i-2)*(2^(k-1)-i),
i=1..min(2^(k-1)-1, n-k+1)), k=1..n):
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MATHEMATICA
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a[n_] := Sum[Sum[Binomial[n-k-1, i-2]*(2^(k-1)-i), {i, 1, Min[2^(k-1) - 1, n - k + 1]}], {k, 1, n}]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Feb 21 2017, after Alois P. Heinz *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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