cp's OEIS Frontend

A211362 lists the binary interpretations of inversion sets ordered by the reverse colexicographic order of permutations (A055089). This permutation orders them by value. Its inverse begins: 0, 1, 3, 2, 4, 5, 12, 13, 8, 6, 9, 7, 16, 14, 17, 15, 10, 11, 18, 19, 21, 20, 22, 23, ...

The finite permutations whose position in reverse colexicographic order is A059590(n) (compare A055089, A195663) have the special feature that their inversion vectors (compare A007623) have only zeros and ones, and give 2*n when interpreted as binary numbers. As the inversion vectors are special, one may also take a look at the inversion sets. This sequence shows them, interpreted as binary numbers (compare A211362).

Index numbers (compare A055089) of transpositions.

Since A057113 is generated based on A057112(n) and successive transpositions, we can be assured that A141826(n) will always map (using A055089)to permutation with an even number of transpositions (applicable, e.g. to the Icosahedron and truncated icosahedron studied in the Geometry Center at http://www.scienceu.com/geometry/facts/solids/handson.html ).

All rows of this array are infinite permutations of the nonnegative integers. Row m (counted from 0) is always generated by modifying the sequence of nonnegative integers in the following way: The sequence of integers is written in reverse binary. Than the finite permutation p_m (row m of array A055089) is applied on the digits of all entries.

The rows of the top left n! X 2^n submatrix describe the rotations and reflections of the n-hypercube that preserve the binary digit sums of the vertex numbers. With permutation composition these permutations form the symmetric group S_n.

Applying such a permutation on the binary string of a Boolean function gives the string of a function in the same big equivalence class (compare A227723).

Triangle row m has length 2^n for m in the interval [(n-1)!,n![. The rest of the array row repeats the same pattern. The first digit of the rest is the digit before plus one.

Index numbers (compare A055089) of permutations like (2,4,6,...,1,3,5...).

Index numbers (compare A055089) of permutations like (n,1, n+1,2, n+2,3, ...).

Index numbers (compare A055089) of chains of transpositions.

Index numbers (compare A055089) of rows of adjacent transpositions.

Every row of this triangle represents a number which is normal in base b >= 2. Since there are infinitely many bases, every row represents a set of infinitely many numbers, each of which is normal in at least one base.

Treating A(n,k) as equal to k when k is greater than A084558(n) (A084558(n) is the length of the n-th row minus one), for any base b >= 2, the concatenation of the base-b expansion of the n-th row is normal in base b. In other words, for any n >= 0 and b >= 2, C(n,b) is normal in base b.

C(n,b) = Sum_{k >= 0} A(n,k)/(b^(Sum_{m=0..k} ceiling(log_b(A(n,m)+1))))

(The equation for Champernowne's constants using the n-th row of this triangle rather than 0,1,2,...)

C(0,b) is Champernowne's constant for base b (C_b).

Even though for any b and m >= 2, b != m, C(n,b) != C(n,m), it is possible for C(n,b) = C(m,p) where n != m and b != p. In such a case, C(n,b) is normal in two different bases. m will likely be significantly larger than n and p will likely be a power of b.