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User:Tilman Piesk

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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.

Boolean functions

A001317( A089633 ) = A211344   Atomic Boolean functions

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

A227725 (A227724)   number of n-ary secs that contain 2^k functions (and the same only for half full functions)

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

A198260   runs of ones,   A227961   corresponding tabf

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

A182176   number of all Boolean functions related to sona of size 2^n. A182176(4) = 307 different Boolean functions can be seen in these 67 sec matrices.


A055089   Finite permutations in reverse colexicographic order (Row n shows relevant digits of the n-th finite permutation.)
A195665   Bit-permutations of non-negative integers

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.)

A083355(n) is the number of preferential arrangements (PA) of partitions of an n-set.
A232598(n,k) is the number of PA with k blocks. A233357(n,k) is the number of PA with k levels.

A187783(m,n) is the number of permutations of a multiset that contains m times the elements of a n-set.
A248814 is its column 6. A248827 shows the row sums.

Arrays of permutations

Odd evening, parted
Odd evening, alternating
Chains of transpositions
Rows of transpositions
Nested transpositions
Circular shift to the right
Circular shift to the left

Partition related number triangles

tabl triangles,
columns match number of blocks:
All Noncr
All A008277
Utr A152175 A209805
Utr&r A152176 A209612
tabl triangles,
columns match number of singletons:
All Noncr
All A124323
Utr A211356 A211357
Utr&r A211358 A211359
tabf triangles,
columns match type:
All Noncr     Rightcol
All A211350
Utr A211352 A211353 A047996
Utr&r A211354 A211355 A052307
Row sums:
All Noncr
All A000110
Utr A084423 A054357
Utr&r A084708 A111275

Walsh permutations

A002884   number of n-bit Walsh permutations
A053601   number of compression vectors with different elements, so A053601(n) = A002884(n) / n!

A195467 (A197819)   Array of Gray code permutation powers (mod 2)

A239303   Compression vectors of square roots of Gray * bit-reversal
A239304   Permutations corresponding to graphs corresponding to A239303

Nimber multiplication and powers of 2

Mat(m,n) = List ( KeyMat(m,n) )

Compressed table of nim-products (A051775)
A223537(m,n) = A223539( A223538(m,n) )

Table of nim-products of powers of 2
A223541(m,n) = A223543( A223542(m,n) )
A223541 is symmetrical. Its lower triangle is A223540.

A002487(N+)=1,  1,2,  1,3,2,3,  1,4,3,5,2,5,3,4...   Stern's diatomic series (probably the number of distinct entries in the antidiagonals of A223541)


A001349   Number of connected graphs with n nodes (


A000217(N+) = 1,3,6,10,15,21...   Triangular numbers
A000292(N+) = 1,4,10,20,35,56...   Tetrahedral numbers

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

A001317(N0) = 1,3,5,15,17,51,85,255...   Sierpinski triangle rows read like binary numbers
A197818(N0) = 1,3,5,15,17,51,93,255...   Antidiagonals of the negated binary Walsh matrix read as binary numbers

A228539 (A228540)   Rows of (negated) binary Walsh matrices read as binary numbers
Most entries in these two sequences are divisible by Fermat numbers (A000215).

A006046(2^n) = 3^n.   Partial sum of Gould's sequence A001316.

A001222 (A001221)   number of (distinct) prime factors of n

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.

The array A000027 n 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12
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


Sierpinski triangles in plots

Very incomplete list of sequences that somehow show a Sierpinski triangle in their scatterplot:

A117966/graph   write n in ternary and then replace 2's with (-1)'s
A227963/graph   sona-secs

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)