A141599 -id:A141599 - cp's OEIS frontend

A141598 Total number of all-interval rows for systems with 2n notes in the octave (2n-edo).

2, 8, 24, 192, 2880, 46272, 1250592, 44095488, 1865756160

Offset: 1

Milan Gustar (artech(AT)noise.cz), Sep 03 2008

This sequence and A141599 are based on the idea of "all-interval rows" from a musical techniques called dodecaphony and serialism.
Twelve tones from an octave (c, c#, d, d#, e, f, ..., b) are marked by numbers 0, 1, ..., 11 (c is 0, c# is 1, etc.).
The "interval" between two notes n1 and n2 is calculated as the difference n2-n1 modulo 12. For example, if note 1 is c# (1) and note 2 is f (5) the interval is 5-1=4, interval between 5 and 1 is 8 (1-5=-4, -4=8 mod 12), etc.
The "all-interval" row is any sequence of twelve notes containing all notes of an octave (0..11) and all intervals (1..11) between adjacent positions. For example, the row 0 1 3 2 7 10 8 4 11 5 9 6 has intervals 1 2 11 5 3 10 8 7 6 4 9, i.e., it is an all-interval row.
There are 46272 such rows from all possible 479001600 (12!) permutations.
Rows with the same interval structure are equivalent in dodecaphony, for example the rows 0, 1, ..., 10, 11 and 1, 2, ..., 11, 0 both have the same intervals (all 1s), the second row is only transposed (moved) one step higher. There are 12 possible transpositions of one row, therefore there are 3856 (46272/12) "non-equivalent" unique all-interval rows.
My generalization is an extension of this principle to microtonal systems - equal divisions of octave, EDO. Rows can be constructed for the tuning systems with any number of notes in the octave, not only 12. As it can be easily proved, the all-interval rows exist only in systems with even number of notes in the octave.
Also the number of permutations of 1..2n which have distinct differences [Gilbert]. - N. J. A. Sloane, Mar 15 2014

E. N. Gilbert, Latin squares which contain no repeated digrams, SIAM Rev. 7 1965 189--198. MR0179095 (31 #3346). Mentions this sequence (see Q(n)). - _N. J. A. Sloane_, Mar 15 2014
Milan Gustar, More information
Milan Gustar, Programs

Cf. A067601, A141599, A155914.

a(n) = 2*n*A141599(n). - Leo C. Stein, Nov 26 2016

a(9) is calculated from A141599(9) after David V. Feldman. - Jud McCranie, Oct 07 2018

A067601 a(n) is the number of inequivalent permutations of {0..2n-1}, such that the first differences (modulo 2n) are a permutation of {1..2n-1}.

Original entry on oeis.org

1, 1, 2, 12, 144, 1928, 44664, 1377984, 51826560

Offset: 1

Eugene McDonnell (eemcd(AT)aol.com), Jan 31 2002

"Inequivalent" effectively means that the permutation begins with 0 and the second item is <= n. (Working modulo 2n, s1+k,s2+k,s3+k,... is equivalent to s1,s2,s3,...; and -s1,-s2,-s3 is equivalent to s1,s2,s3,...)

The references all deal with length 12.

			0 1 3 2 has first difference, mod 4, of 1 2 3;
0 2 1 4 5 3 has first difference, mod 6, of 2 5 3 1 4;
0 4 5 8 3 1 7 9 2 11 10 6 has first difference, mod 12, of 4 1 3 7 10 6 2 5 9 11 8.

Stefan Bauer-Mengelberg and Melvin Ferentz, On Eleven-Interval Twelve-Tone Rows, Perspectives of New Music 3, no. 2 (Spring-Summer 1965): 93-103
Sean A. Irvine, Java program (github)
Robert Morris and Daniel Starr, The Structure of All-interval Series, Journal of Music Theory 18, no. 2 (Fall 1974): 364-389.
David Schiff, Elliott Carter's Harvest Home, Tempo 167 (December 1988): 7-13.

Cf. A141598, A141599.

a(n) = ceiling(A141599(n)/2). - Leo C. Stein, Nov 26 2016

Edited by Don Reble, Oct 31 2005

a(9) from Sean A. Irvine, Dec 22 2023

A155914 Example of an all interval series: the 12 integers 0..11 sorted such that the first differences contain all numbers from 1 to 11 (mod 12).

Original entry on oeis.org

0, 11, 7, 4, 2, 9, 3, 8, 10, 1, 5, 6

Offset: 1

Craig Bourne (cbourne(AT)cbourne.com), Jan 30 2009

"All-interval" means that the differences 11-0=11, 7-11=-4, 4-7=-3, ..., 6-5=1 read modulo 12 contain all numbers (intervals) from 1 to 11.
This is one of 3856 such sequences.
The Anders link contains a source program written in Strasheela, formulated as a constrained satisfaction problem (CSP).

Robert Morris and Daniel Starr, The Structure of All-Interval Series, 1974, Yale University Department of Music.

Torsten Anders, All Interval Series, (2009).
Milan Gustar, More information
Milan Gustar, Programs and data

Cf. A141598, A141599.

A238838 Number of 2n X 2n addition squares in which every digram (s,s;), s' != s, appears once horizontally and once vertically.

Original entry on oeis.org

2, 48, 5760, 5806080, 75246796800, 1780537083494400, 115939740156316876800, 19864514173849162481664000

Offset: 1

N. J. A. Sloane, Mar 15 2014

Gilbert, E. N., Latin squares which contain no repeated digrams, SIAM Rev. 7 1965 189--198. MR0179095 (31 #3346).

Cf. A000315, A002860, A000010, A141599.

a(n) = (2*n)!*A141599(n)^2/phi(n).

A322253 Number of circular permutations of the integers from 0 to n which generate a complete stepping-on sequence, when the stepping-on direction depends on the odd/even parity of the current value.

Original entry on oeis.org

1, 2, 2, 4, 4, 48, 24, 32, 288, 3072, 3856, 38272, 89328, 1294080

Offset: 1

Ian Duff, Dec 01 2018

There are n! circular permutations of the integers from 0 to n. Only some have the property that a complete sequence of these integers can be found by choosing a start value then continuing to move left or right by the number of steps indicated, the direction of each move depending on whether the current value is odd or even. For example, of the six permutations for n = 3, only 0132 and 0213 generate a complete sequence if an odd value gives a leftward move that number of places and an even value gives a rightward move that number of places. If the direction rule is reversed, the two valid permutations are 0231 and 0312, the reverse of the previous two. Thus a(3) = 2.
Putting 0 in position 1 at the left and counting rightwards, the starting position for a complete sequence is 2 + floor(n/2) for the rule odd/left, even/right and floor((n+3)/2) for the opposite rule. As a further example, one of the a(9) = 288 valid permutations using the former rule for n = 9 is 0986423175. Starting at position 2 + floor(9/2), i.e., 6, the sequence 2, 1, 3, 6, 5, 4, 7, 9, 8, 0 is found. Clearly, all such sequences end with zero.
It is conjectured that the sequence continues indefinitely.
From David A. Corneth, Dec 05 2018: (Start)
For even n, if [d1, d2, ..., dn] is a valid permutation then so is [n + 1 - d1, n + 1 - d2, ..., n + 1 - dn] which is a different permutation.
More generally, for any valid permutation where n is even, di and n+1 - di can be interchanged for any value (where n + 1 - di != di) to give another valid permutation. Hence a(n) is divisibly by 2^k for n = 2*k.
a(n) > 0. For n = 1, [0, 1] is valid, for n = 2, [0, 1, 2] is valid; for n = 3, [0, 3, 1, 2] is valid etc. By taking this valid tuple from n by adding n + 1 to the right of the permutation or to the right of the 0, depending on the parity of n, one finds another valid tuple. Hence a(n) > 0. (End)
For the complementary case where stepping-on is always in the same direction, no permutation of 0 to n with n even can generate a complete sequence. For odd n, the number of complete sequences corresponds to A141599((n + 1)/2) for n up to 11, as limited by available computing power - it is conjectured that this correspondence continues indefinitely. - Ian Duff, Dec 25 2018

David A. Corneth, tuples for a(n) for n = 1..12
David A. Corneth, PARI program

a(2n - 1) = A141599(n) for n up to 7, again limited by available computing power. It is conjectured that the correspondence continues indefinitely.

\\ See PARI link \\ David A. Corneth, Dec 04 2018

a(12)-a(14) from David A. Corneth, Dec 04 2018

A364237 a(n) is the number of non-equivalent permutations of {1,2,...,2n-1} such that no subset of consecutive terms from the permutation sums to 0 modulo 2n, where two permutations are equivalent if one can be obtained from the other by multiplying every entry with an integer relatively prime to 2n and/or reversing the permutation.

Original entry on oeis.org

1, 1, 2, 4, 42, 504, 7492, 172480, 8639632

Offset: 1

Anna Coleman, Joshua Harrington, Maggie X. Lai, Philip Thomas, and Wing Hong Tony Wong, Jul 18 2023

If we consider all permutations of {1,2,...,2n-1} such that no subset of consecutive terms from the permutation sums to 0 modulo 2n, then the number of such permutations is given by the number of constructive orderings mentioned in A141599. For example, given the permutation 14325 that satisfies the given conditions, observe that the partial sums modulo 6, namely 1=1, 1+4=5, 1+4+3=2, 1+4+3+2=4, and 1+4+3+2+5=3, are distinct.

			When n=3, there are four permutations of {1,2,3,4,5} such that no subset of consecutive terms from the permutation sums to 0 modulo 6, namely 14325, 25314, 41352, and 52341. Note that 14325 and 52341 are equivalent by reversing the permutations. Furthermore multiplication by 5 on every entry also yields the same equivalence. Additionally, 25314 and 41352 are analogously equivalent. Hence a(3)=2.
When n=4, 6142573 and 3752416 are equivalent by reversing the permutations but not by multiplying any integer relatively prime to 8, whereas 6142573 and 2346751 are equivalent by multiplication of 3 on every entry.

Sunil K. Chebolu and Papa A. Sissokho, Zero-sum-free tuples and hyperplane arrangements, Integers 22 (2022), #A13.
Sean A. Irvine, Java program (github)

Cf. A141599.

n = 3 #the index for the sequence a(n)
orbits = {} #dictionary of permutations that are consecutive zero-sum-free
seen = [] #list of seen permutations that are consecutive zero-sum-free
a = 0 #the value of a(n)
for labeling in Permutations(range(1,2*n)):
    if labeling not in seen:
        sums = [labeling[0]]
        for i in range(1,2*n-1):
            nextsum = (labeling[i] + sums[i-1]) % (2*n)
            if any([nextsum == 0, nextsum in sums]):
                break
            sums.append(nextsum)
        if len(sums) == (2*n)-1:
            a += 1
            orbits[a] = []
            for m in [x for x in range(1,2*n) if gcd(x,2*n) == 1]:
                equiv = [(m*labeling[i]) % (2*n) for i in range(2*n-1)]
                if equiv not in orbits[a]:
                    orbits[a].append(equiv)
                seen.append(equiv)
                equiv = [equiv[2*n-2-i] for i in range(2*n-1)]
                if equiv not in orbits[a]:
                    orbits[a].append(equiv)
                seen.append(equiv)
print(f"a({n}) = {a}\n")
print("Equivalencies:")
for i in range(1,a+1):
    print(f"{i}.")
    for x in orbits[i]:
        print(x)
    print('\n')

a(8)-a(9) from Sean A. Irvine, Aug 15 2023

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A141598 Total number of all-interval rows for systems with 2n notes in the octave (2n-edo).

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A067601 a(n) is the number of inequivalent permutations of {0..2n-1}, such that the first differences (modulo 2n) are a permutation of {1..2n-1}.

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A155914 Example of an all interval series: the 12 integers 0..11 sorted such that the first differences contain all numbers from 1 to 11 (mod 12).

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A238838 Number of 2n X 2n addition squares in which every digram (s,s;), s' != s, appears once horizontally and once vertically.

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A322253 Number of circular permutations of the integers from 0 to n which generate a complete stepping-on sequence, when the stepping-on direction depends on the odd/even parity of the current value.

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