A056413 -id:A056413 - cp's OEIS frontend

A285548 Array read by antidiagonals: T(m,n) = number of step cyclic shifted sequences of length n using a maximum of m different symbols.

1, 1, 2, 1, 3, 3, 1, 4, 6, 4, 1, 6, 10, 10, 5, 1, 6, 21, 20, 15, 6, 1, 13, 24, 55, 35, 21, 7, 1, 10, 92, 76, 120, 56, 28, 8, 1, 24, 78, 430, 201, 231, 84, 36, 9, 1, 22, 327, 460, 1505, 462, 406, 120, 45, 10, 1, 45, 443, 2605, 2015, 4291, 952, 666, 165, 55, 11

Offset: 1

Andrew Howroyd, Apr 20 2017

See A056371, A002729 for an explanation of step shifts. Under step cyclic shifts, abcde, bdace, bcdea, cdeab and daceb etc. are equivalent.
Equivalently, the number of mappings with domain {0..n-1} and codomain {1..m} up to equivalence.  Mappings A and B are equivalent if there is a d, prime to n, and a t such that A(i) = B((i*d + t) mod n) for i in {0..n-1}.
All column sequences are polynomials of order n.

			Table starts:
1  1  1   1   1     1     1      1      1       1 ...
2  3  4   6   6    13    10     24     22      45 ...
3  6 10  21  24    92    78    327    443    1632 ...
4 10 20  55  76   430   460   2605   5164   26962 ...
5 15 35 120 201  1505  2015  14070  37085  246753 ...
6 21 56 231 462  4291  6966  57561 188866 1519035 ...
7 28 84 406 952 10528 20140 192094 752087 7079800 ...
...

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Andrew Howroyd, Table of n, a(n) for n = 1..1275
R. C. Titsworth, Equivalence classes of periodic sequences, Illinois J. Math., 8 (1964), 266-270.

Rows 2-6 are A002729, A056411, A056412, A056413, A056414.

Cf. A285522, A132191.

IsLeastPoint[s_, f_] := Module[{t=f[s]}, While[t>s, t=f[t]]; Boole[s==t]];
c[n_, k_, t_] := Sum[IsLeastPoint[u, Mod[#*k+t, n]&], {u, 0, n-1}];
a[n_, x_] := Sum[If[GCD[k, n] == 1, x^c[n, k, t], 0], {t, 0, n-1}, {k, 1,
n}] / (n*EulerPhi[n]);
Table[a[n-m+1, m], {n, 1, 11}, {m, 1, n}] // Flatten (* Jean-François Alcover, Jun 05 2017, translated from PARI *)

IsLeastPoint(s,f)={my(t=f(s)); while(t>s,t=f(t));s==t}
C(n,k,t)=sum(u=0,n-1,IsLeastPoint(u,v->(v*k+t)%n));
a(n,x)=sum(t=0, n-1, sum(k=1, n, if (gcd(k, n)==1, x^C(n,k,t),0)))/(n * eulerphi(n));
for(m=1, 7, for(n=1, 10, print1( a(n,m), ", ") ); print(); );

A056411 Number of step cyclic shifted sequences using a maximum of three different symbols.

Original entry on oeis.org

3, 6, 10, 21, 24, 92, 78, 327, 443, 1632, 1698, 12769, 10464, 57840, 122822, 348222, 476052, 3597442, 3401970, 22006959, 41597374, 142677588, 186077886, 1476697627, 1694658003, 8147282460, 15690973754, 68149816689, 84520682160, 857935531804, 664166389302, 3620293575942, 8422974597554, 30656600391720, 59561470990362

Offset: 1

Marks R. Nester

nonn

See A056371 for an explanation of step shifts. Under step cyclic shifts, abcde, bdace, bcdea, cdeab and daceb etc. are equivalent.

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

D. Z. Dokovic, I. Kotsireas et al., Charm bracelets and their application to the construction of periodic Golay pairs, arXiv:1405.7328 [math.CO], 2014.
R. C. Titsworth, Equivalence classes of periodic sequences, Illinois J. Math., 8 (1964), 266-270.

Row 3 of A285548.
Cf. A002729.
Cf. A056412, A056413, A056414.

M[j_, L_] := Module[{m=1}, While[Sum[j^i, {i, 0, m-1}] ~Mod~ L != 0, m++]; m]; c[j_, t_, n_] := Sum[1/M[j, n/GCD[n, u*(j-1)+t]], {u, 0, n-1}]; CB[n_, k_] = If [n==1, k, 1/(n*EulerPhi[n])*Sum[If[1==GCD[n, j], k^c[j, t, n], 0], {t, 0, n-1}, {j, 1, n-1}]]; Table[Print[cb = CB[n, 3]]; cb, {n, 1, 35}] (* Jean-François Alcover, Dec 04 2015, after Joerg Arndt *)

\\ see p.3 of the Dokovic et al. reference
M(j,  L)={my(m=1); while ( sum(i=0, m-1, j^i) % L != 0, m+=1 ); m; }
c(j, t, n)=sum(u=0,n-1, 1/M(j, n / gcd(n, u*(j-1)+t) ) );
CB(n, k)=if (n==1,k, 1/(n*eulerphi(n)) * sum(t=0,n-1, sum(j=1,n-1, if(1==gcd(n,j), k^c(j,t,n), 0) ) ) );
for(n=1, 66, print1(CB(n,3),", "));
\\ second argument k=3, 4, 5, 6 respectively gives A056411, A056412, A056413, A056414.
\\ Joerg Arndt, Aug 27 2014

Refer to Titsworth or slight "simplification" in Nester.

Added more terms, Joerg Arndt, Aug 27 2014

A056374 Number of step shifted (decimated) sequences using a maximum of five different symbols.

Original entry on oeis.org

5, 25, 75, 375, 825, 8125, 13175, 103125, 327125, 2445625, 4884435, 61640625, 101732425, 1017323125, 3816215625, 19104609375, 47683838325, 635787765625, 1059638680675, 11924780390625, 39736963221875, 238418603522125, 541860418146375

Offset: 1

Marks R. Nester

nonn

See A056371 for an explanation of step shifts.

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

G. C. Greubel, Table of n, a(n) for n = 1..1000
R. C. Titsworth, Equivalence classes of periodic sequences, Illinois J. Math., 8 (1964), 266-270.

Cf. A056413.

A row or column of A132191.

a[m_, n_] := (1/EulerPhi[n])*Sum[If[GCD[k, n] == 1, m^DivisorSum[n, EulerPhi[#]/MultiplicativeOrder[k, #] &], 0], {k, 1, n}]; Table[a[5, n], {n, 1, 23}] (* Jean-François Alcover, Dec 04 2015 *)

The cycle index is implicit in Titsworth.

Sequences A056372-A056375 fit a general formula, implemented in PARI/GP as follows: { a(m,n) = sum(k=1, n, if(gcd(k, n)==1, m^sumdiv(n, d, eulerphi(d)/znorder(Mod(k, d))), 0); ) / eulerphi(n) }. - Max Alekseyev, Nov 08 2007

More terms from Max Alekseyev, Nov 08 2007

A056432 Step cyclic shifted sequence structures using a maximum of five different symbols.

Original entry on oeis.org

1, 2, 3, 7, 8, 36, 35, 201, 393, 2370, 3971, 46094, 66675, 613439, 2139350, 10016839, 23424253, 294723254, 465082931, 4972761822, 15773188813, 90323063945, 196342666487, 2587522663248, 4967200045397, 39800811900389

Offset: 1

Marks R. Nester

nonn

See A056371 for an explanation of step shifts. Under step cyclic shifts, abcde, bdace, bcdea, cdeab and daceb etc. are equivalent. Permuting the symbols will not change the structure.

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Cf. A056413.

Use de Bruijn's generalization of Polya's enumeration theorem as discussed in reference.

A056418 Number of step cyclic shifted sequences using exactly five different symbols.

Original entry on oeis.org

0, 0, 0, 0, 6, 150, 400, 4080, 15480, 127818, 269340, 3493680, 5777190, 57262450, 210945182, 1030388115, 2493913170, 32176448060, 51785999300, 562228325904, 1805427491920, 10438821843750, 22865672706000

Offset: 1

Marks R. Nester

nonn

See A056371 for an explanation of step shifts. Under step cyclic shifts, abcde, bdace, bcdea, cdeab and daceb etc. are equivalent.

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Cf. A056413.

A056413(n)-5*A056412(n)+10*A056411(n)-10*A002729(n)+5.

A056419 Number of step cyclic shifted sequences using exactly six different symbols.

Original entry on oeis.org

0, 0, 0, 0, 0, 60, 360, 6030, 35280, 410976, 1174896, 19948200, 43022700, 543757860, 2524673904, 15399118440, 46338868800, 737917466160, 1459120076400, 19381180990752, 75959665269840, 534453892557660

Offset: 1

Marks R. Nester

nonn

See A056371 for an explanation of step shifts. Under step cyclic shifts, abcde, bdace, bcdea, cdeab and daceb etc. are equivalent.

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Cf. A056414.

A056414(n)-6*A056413(n)+15*A056412(n)-20*A056411(n)+15*A002729(n)-6.

A056422 Number of primitive (period n) step cyclic shifted sequences using a maximum of five different symbols.

Original entry on oeis.org

5, 10, 30, 105, 196, 1460, 2010, 13950, 37050, 246542, 445510, 5204180, 7832180, 72701620, 254689426, 1196199375, 2805046960, 35322773200, 55770979190, 596439488166, 1892294576710, 10837222569140

Offset: 1

Marks R. Nester

nonn

See A056371 for an explanation of step shifts. Under step cyclic shifts, abcde, bdace, bcdea, cdeab and daceb etc. are equivalent.

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Sum mu(d)*A056413(n/d) where d|n.

cp's OEIS Frontend

A285548 Array read by antidiagonals: T(m,n) = number of step cyclic shifted sequences of length n using a maximum of m different symbols.

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A056411 Number of step cyclic shifted sequences using a maximum of three different symbols.

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A056374 Number of step shifted (decimated) sequences using a maximum of five different symbols.

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A056432 Step cyclic shifted sequence structures using a maximum of five different symbols.

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A056418 Number of step cyclic shifted sequences using exactly five different symbols.

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A056419 Number of step cyclic shifted sequences using exactly six different symbols.

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A056422 Number of primitive (period n) step cyclic shifted sequences using a maximum of five different symbols.

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