querying multidimensional sequences in OEIS

[franktaw at netscape.net]

<Me>
>Off hand, I can't think of any interesting functions of finite 
sequences of
>positive integers that depend on the order of the sequence.

Of course, I no sooner wrote that than I started thinking of such 
functions.
I'm not sure any of these are tremendously interesting, but they're at 
least
somewhat so.

In the following, b(i), i = 1..k is the finite sequence.  I'm starting 
all these
sequences with the empty sequence, but I searched for them without
that first term.  All are shown here in A066099 order.

Alternating sum
Sum_i (-1)^{i+1) b(i)
0,1,2,0,3,1,-1,1,4,2,0,2,-2,0,2,0

Binomial sum
Sum_i C(k-1,i-1) b(i)
0,1,2,2,3,3,3,4,4,4,4,5,4,6,5,8

Inverse binomial sum
Sum_i (-1)^{i+1) C(k-1,i-1) b(i)
0,1,2,0,3,1,-1,0,4,2,0,1,-2,-2,1,0

Weighted sum
sum_i i b(i)
0,1,2,3,3,4,5,6,4,5,6,7,7,8,9,10
This is A029931

Zero-based weighted sum
sum_i (i-1) b(i)
0,0,0,1,0,1,2,3,0,1,2,3,3,4,5,6

Sum of products of consecutive elements
sum_{i=1}^{k-1} b(i) b(i+1)
0,0,0,1,0,2,2,2,0,3,4,3,3,4,3,3

Number of rises
sum_{b(i)>b(i-1)} 1
0,0,0,0,0,0,1,0,0,0,0,0,1,1,1,0

Number of falls
sum_{b(i)<b(i-1)} 1
0,0,0,0,0,1,0,0,0,1,0,1,0,1,0,0

Number of unchanged
sum_{b(i)=b(i-1)} 1
0,0,0,1,0,0,0,2,0,0,1,1,0,0,1,3

Number of non-rises
sum_{b(i)<=b(i-1)} 1
0,0,0,1,0,0,1,2,0,0,1,1,1,1,2,3

Number of non-falls
sum_{b(i)>=b(i-1)} 1
0,0,0,1,0,1,0,2,0,1,1,2,0,1,1,3

Number of monotonically increasing runs
1 + number of falls, but 0 for empty sequence
0,1,1,1,1,2,1,1,1,2,1,2,1,2,1,1

Number of monotonically decreasing runs
1 + number of rises, but 0 for empty sequence
0,1,1,1,1,1,2,1,1,1,1,1,2,2,2,1

Number of runs of equal terms
1 + number of unchanged, but 0 for empty sequence
0,1,1,2,1,1,1,3,1,1,2,2,1,1,2,4

Number of distinct non-empty subsequences
0,1,1,2,1,3,3,3,1,3,2,5,3,5,5,4
E.g., f(1,1) = 2, for the sequences [1], and [1,1].

Number of distinct subsequences
1 + number of distinct non-empty subsequences
1,2,2,3,2,4,4,4,2,4,3,6,4,6,6,5

Number of set partitions
List parts of set partition by their smallest element, and count the 
part sizes.
E.g., 1|2,4|3 (also known as {1,2,3,2}) would count for the sequence 
[1,2,1].
1,1,1,1,1,2,1,1,1,3,3,3,1,2,1,1

Number of permutations
List permutations in cycle form, sorted by the smallest element in the 
cycle,
and count the cycle lengths.
Number of set partitions * Product_i (b(i)-1)!
1,1,1,1,2,2,1,1,6,6,3,3,2,2,1,1

Number of partially ordered sets (unlabelled) by rank
The rank of an element in a poset is the length of the longest chain of 
which it
is the largest element.
1,1,1,1,1,2,1,1,1,3,4,3,1,2,1,1

Number of partially ordered sets (labelled) by rank
1,1,1,2,1,9,3,6,1,28,54,60,4,36,12,24

Number of partially ordered sets (naturally labelled) by rank
Naturally labelled means labels are consistent with the ordering.
1,1,1,1,1,4,1,1,1,11,13,8,1,4,1,1

Number of forests (unlabelled, unordered) with b(i) nodes at height i.
1,1,1,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,2,1,2,2,1,1,1,1,2,1,1,1,1,1

... and, of course, various other forest options.

If there is agreement that this is the right order for these sequences, 
I'll try
to find time to submit them.
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