I am a 1982 born mathematical autodidact interested in discrete mathematics.
I contribute to various Wikimedia projects under the name Watchduck.
My native language is German, and I live in Augsburg.
This page shows some sequences I am interested in. My own ones have this background. A list of my sequences is here.
- 1 Boolean functions
- 2 Combinatorics
- 3 Graphs
- 4 Misc.
- 5 Sierpinski triangles in plots
A227722, A227723 Smallest elements in equivalence classes of Boolean functions (sec, bec), the latter a subsequence of the former
A054724, A039754 number of secs, becs of n-ary Boolean functions by weight
A000231, A000616 number of secs, becs of n-ary Boolean functions, row sums of the two triangles above
Subgroups of nimber addition (sona)
A190939 sona interpreted as binary numbers (my first sequence, added on 2011-05-24)
The 2-binomial coefficients give their number by weight: A022166(n,k) is the number of sona of size 2^n with weight 2^k. A022166(4,0...4) = (1,15,35,15,1).
A006116(n) is the number of all sona of size 2^n. A006116(n) = 67.
A227963 sona-secs (entries are from A227722) The same secs in the same order as in A190939, but represented by the smallest among the numeric values of its functions, instead of the unique odd one
A227960 sona-becs (subsequence of A227723)
A076831(n,k) is the number of becs of sona of size 2^n with weight 2^k. A076831(4,0...4) = (1,4,6,4,1). Not to be confused with Pascal's triangle (A007318).
A076766(n) is the number of all becs of sona of size 2^n. A076766(4) = 16. Not to be confused with powers of two (A000079).
A227962 Permutations that assign complementary sona-becs to each other
A000041(n) is the number of integer partitions of n
A194602 Integer partitions interpreted as binary numbers
A000110(n) is the number of partitions of an n-set (Bell numbers)
A231428 Set partitions interpreted as binary numbers
A211362 (A211363) Inversion sets of finite permutations interpreted as binary numbers (and the corresponding permutation of the integers)
A059590(n)-th finite permutations have inversion vectors (A007623) that look like n in binary. (Or 2*n when a useless zero is appended.)
A211362( A059590 ) = A211364 shows the corresponding inversion sets interpreted as binary numbers.
Rencontres numbers: A008290(n,k) among first n! finite permutations leave k elements unchanged. Left column (k=0) shows numbers of derangements (A000166).
Refined rencontres numbers: A181897(n,k) among the first n! finite permutations have cycle type k. (See refined r.n. under reflected r.n.)
A198380 Cycle type (i.e. integer partition) of n-th finite permutation, represented by index number of A194602
A000629(N+) = 2,6,26,150,1082... Necklaces of partitions of n+1 labeled beads. Logically distinct strings of first order quantifiers with n variables.
A000670(N+) = 1,3,13, 75, 541... Ordered Bell numbers counting ordered set partitions (half of last sequence)
A019538(n,k) = k!*A008277(n,k) is the number of (n-k)-faces of the permutohedron of order n, and thus the number of ordered set partitions with n elements. (Row sums are the ordered Bell numbers.)
Odd evening, parted
Odd evening, alternating
Chains of transpositions
Rows of transpositions
Circular shift to the right
Circular shift to the left
columns match number of blocks:
columns match number of singletons:
columns match type:
Mat(m,n) = List ( KeyMat(m,n) )
A018900(N+) = 3, 5,6, 9,10,12, 17,18,20,24... Numbers that contain 2 binary ones
A014311(N+) = 7, 11, 13, 14, 19, 21, 22, 25, 26, 28, 35, 37, 38, 41, 42, 44, 49, 50, 52, 56... Numbers that contain 3 binary ones
A001055 multiplicative partitions
And my 2 cents about lists with offset 0.
The array of positive integers
T(m,n) = m + (m+n-2)(m+n-1)/2
The rows, columns and diagonals of this array are important because they can be used to calculate the rows, columns and diagonals of other arrays.
|Main diagonal||A001844||2n^2 + 2n + 1||1,||5,||13,||25,||41,||61,||85,||113,||145,||181,||221,||265|
|First knight moves diagonal||A064225||1,||8,||24,||49,||83,||126,||178,||239,||309,||388,||476,||573|
|Second knight moves diagonal||A081267||( 9n^2 + 7n + 2 ) / 2||1,||9,||26,||52,||87,||131,||184,||246,||317,||397,||486,||584|
|col 1||A000217||( n^2 + n ) / 2||1,||3,||6,||10,||15,||21,||28,||36,||45,||55,||66,||78|
|col 2||A000096||( n^2 + 3n ) / 2||2,||5,||9,||14,||20,||27,||35,||44,||54,||65,||77,||90|
- Infinite enumeration of finite partitions (2012-02-02) Are there generally accepted orderings of all finite integer partitions or set partitions?
- Stern's diatomic series, n-phi-torial and the primes (2013-04-10) The most numerous entries in the rows of Stern's diatomic series (A002487) all seem to be from A193339 - and thus to be prime numbers.
- Definition of A211351
Sierpinski triangles in plots
Very incomplete list of sequences that somehow show a Sierpinski triangle in their scatterplot:
A080099/graph (n AND k), A080098/graph (n OR k), A051933/graph (n XOR k), A003987/graph (n XOR m, symmetric nimber addition table)
A223541/graph (A223540/graph, A223542/graph) nim-products of powers of 2 (lower triangle, key-matrix)