A228820 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).

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

Offset: 0

Author

Omar E. Pol, Sep 05 2013

Keywords

• nonn

Comments

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.

From Omar E. Pol, Feb 07 2014: (Start)

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)

Examples

			Table 1. Compositions of 4 in lexicographic order.
---------------------------------------------------------
j   Composition   Diagram       F-rank          L-rank
---------------------------------------------------------
.                 _ _ _ _
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
.
Table 2. Compositions of 4 in colexicographic order.
---------------------------------------------------------
j   Composition   Diagram       F-rank          L-rank
---------------------------------------------------------
.                 _ _ _ _
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
.
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.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A011782, A001511, A001792, A006519, A195012, A209616, A228369, A228525, A228821.

Programs

Maple

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):
seq(a(n), n=0..50);  # Alois P. Heinz, Sep 09 2013

Mathematica

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 *)

Extensions

More terms from Alois P. Heinz, Sep 09 2013