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A285548
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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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10
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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
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OFFSET
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1,3
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COMMENTS
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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.
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REFERENCES
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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]
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LINKS
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EXAMPLE
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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 ...
...
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MATHEMATICA
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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 *)
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PROG
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(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(); );
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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