A285037
Irregular triangle read by rows: T(n,k) is the number of primitive (period n) periodic palindromic structures using exactly k different symbols, 1 <= k <= n/2 + 1.
Original entry on oeis.org
1, 0, 1, 0, 1, 0, 2, 1, 0, 3, 1, 0, 4, 5, 1, 0, 7, 6, 1, 0, 10, 18, 7, 1, 0, 14, 25, 10, 1, 0, 21, 63, 43, 10, 1, 0, 31, 90, 65, 15, 1, 0, 42, 202, 219, 85, 13, 1, 0, 63, 301, 350, 140, 21, 1, 0, 91, 650, 1058, 618, 154, 17, 1, 0, 123, 965, 1701, 1050, 266, 28, 1
Offset: 1
Triangle starts:
1
0 1
0 1
0 2 1
0 3 1
0 4 5 1
0 7 6 1
0 10 18 7 1
0 14 25 10 1
0 21 63 43 10 1
0 31 90 65 15 1
0 42 202 219 85 13 1
0 63 301 350 140 21 1
0 91 650 1058 618 154 17 1
0 123 965 1701 1050 266 28 1
0 184 2016 4796 4064 1488 258 21 1
0 255 3025 7770 6951 2646 462 36 1
0 371 6220 21094 24914 12857 3222 410 26 1
0 511 9330 34105 42525 22827 5880 750 45 1
...
Example for n=6, k=2:
There are 6 inequivalent solutions to A285012(6,2) which are 001100, 010010, 000100, 001010, 001110, 010101. Of these, 010010 and 010101 have a period less than 6, so T(6,2) = 6-2 = 4.
- 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]
-
\\ Ach is A304972
Ach(n,k=n) = {my(M=matrix(n, k, n, k, n>=k)); for(n=3, n, for(k=2, k, M[n, k]=k*M[n-2, k] + M[n-2, k-1] + if(k>2, M[n-2, k-2]))); M}
T(n,k=n\2+1) = {my(A=Ach(n\2+1,k), S=matrix(n\2+1, k, n, k, stirling(n,k,2))); Mat(vectorv(n, n, sumdiv(n, d, moebius(d)*(S[(n/d+1)\2, ] + S[n/d\2+1, ] + if((n-d)%2, A[(n/d+1)\2, ] + A[n/d\2+1, ]))/if(d%2, 2, 1) )))}
{ my(A=T(20)); for(n=1, matsize(A)[1], print(A[n,1..n\2+1])) } \\ Andrew Howroyd, Oct 01 2019
-
\\ column sequence using above code.
ColSeq(n, k=2) = { Vec(T(n,k)[,k]) } \\ Andrew Howroyd, Oct 01 2019
A309784
T(n,k) is the number of non-equivalent distinguishing coloring partitions of the cycle on n vertices with exactly k parts. Regular triangle read by rows, n >= 1, 1 <= k <= n.
Original entry on oeis.org
0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 4, 2, 1, 0, 1, 8, 10, 3, 1, 0, 1, 25, 32, 16, 3, 1, 0, 4, 62, 129, 84, 27, 4, 1, 0, 7, 176, 468, 433, 171, 37, 4, 1, 0, 18, 470, 1806, 2260, 1248, 338, 54, 5, 1, 0, 31, 1311, 6780, 11515, 8388, 3056, 590, 70, 5, 1, 0, 70, 3620, 25917, 58312, 56065, 26695, 6907, 1014, 96, 6, 1
Offset: 1
The triangle begins:
0;
0, 0;
0, 0, 1;
0, 0, 1, 1;
0, 0, 4, 2, 1;
0, 1, 8, 10, 3, 1;
0, 1, 25, 32, 16, 3, 1;
0, 4, 62, 129, 84, 27, 4, 1;
0, 7, 176, 468, 433, 171, 37, 4, 1;
0, 18, 470, 1806, 2260, 1248, 338, 54, 5, 1;
...
For n=6, we can partition the vertices of C_6 into exactly 3 parts in 8 ways such that all these partitions induce distinguishing colorings for C_6 and that all the 8 partitions are non-equivalent. The partitions are as follows:
{ { 1 }, { 2 }, { 3, 4, 5, 6 } }
{ { 1 }, { 2, 3 }, { 4, 5, 6 } }
{ { 1 }, { 2, 3, 4, 6 }, { 5 } }
{ { 1 }, { 2, 3, 5 }, { 4, 6 } }
{ { 1 }, { 2, 3, 6 }, { 4, 5 } }
{ { 1 }, { 2, 4, 5 }, { 3, 6 } }
{ { 1, 2 }, { 3, 4 }, { 5, 6 } }
{ { 1, 2 }, { 3, 5 }, { 4, 6 } }
For n=6, the above 8 partitions can be written as the following 3 colored bracelet structures: ABCCCC, ABBCCC, ABBBCB, ABBCBC, ABBCCB, ABCBBC, AABBCC, AABCBC. - _Andrew Howroyd_, Sep 22 2019
-
\\ Ach is A304972 and R is A152175 as square matrices.
Ach(n)={my(M=matrix(n, n, i, k, i>=k)); for(i=3, n, for(k=2, n, M[i, k]=k*M[i-2, k] + M[i-2, k-1] + if(k>2, M[i-2, k-2]))); M}
R(n)={Mat(Col([Vecrev(p/y, n) | p<-Vec(intformal(sum(m=1, n, eulerphi(m) * subst(serlaplace(-1 + exp(sumdiv(m, d, y^d*(exp(d*x + O(x*x^(n\m)))-1)/d))), x, x^m))/x))]))}
T(n)={my(A=Ach(n), M=R(n), S=matrix(n, n, n, k, stirling(n, k, 2))); Mat(vectorv(n, n, sumdiv(n, d, moebius(d)*(M[n/d,] + A[n/d,])/2 - moebius(d)*(S[(n/d+1)\2, ] + S[n/d\2+1, ] + if((n-d)%2, A[(n/d+1)\2, ] + A[n/d\2+1, ]))/if(d%2, 2, 1) )))}
{ my(A=T(12)); for(n=1, #A, print(A[n, 1..n])) } \\ Andrew Howroyd, Oct 02 2019
A056503
Number of periodic palindromic structures of length n using a maximum of two different symbols.
Original entry on oeis.org
1, 2, 2, 4, 4, 7, 8, 14, 16, 26, 32, 51, 64, 100, 128, 198, 256, 392, 512, 778, 1024, 1552, 2048, 3091, 4096, 6176, 8192, 12324, 16384, 24640, 32768, 49222, 65536, 98432, 131072, 196744, 262144, 393472, 524288, 786698, 1048576, 1573376, 2097152, 3146256, 4194304
Offset: 1
From _Gus Wiseman_, Sep 16 2018: (Start)
The sequence of palindromic cyclic compositions begins:
(1) (2) (3) (4) (5) (6) (7)
(11) (111) (22) (113) (33) (115)
(112) (122) (114) (133)
(1111) (11111) (222) (223)
(1122) (11113)
(11112) (11212)
(111111) (11122)
(1111111)
(End)
- 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.
A000740,
A000837,
A008965,
A025065,
A059966,
A242414,
A296302,
A317085,
A317086,
A317087,
A318731.
-
(* b = A164090, c = A045674 *)
b[n_] := (1/4)*(7 - (-1)^n)*2^((1/4)*(2*n + (-1)^n - 1));
c[0] = 1; c[n_] := c[n] = If[EvenQ[n], 2^(n/2-1) + c[n/2], 2^((n-1)/2)];
a[n_?OddQ] := b[n]/2; a[n_?EvenQ] := (1/2)*(b[n] + c[n/2]);
Array[a, 45] (* Jean-François Alcover, Oct 08 2017, after Andrew Howroyd *)
Table[Length[Select[Join@@Permutations/@IntegerPartitions[n],Function[q,And[Array[OrderedQ[{q,RotateRight[q,#]}]&,Length[q]-1,1,And],Array[SameQ[RotateRight[q,#],Reverse[RotateRight[q,#]]]&,Length[q],1,Or]]]]],{n,15}] (* Gus Wiseman, Sep 16 2018 *)
A056504
Number of periodic palindromic structures of length n using a maximum of three different symbols.
Original entry on oeis.org
1, 1, 2, 2, 5, 5, 12, 14, 33, 41, 90, 122, 259, 365, 756, 1094, 2233, 3281, 6642, 9842, 19813, 29525, 59292, 88574, 177527, 265721, 532170, 797162, 1595443, 2391485, 4785156, 7174454, 14352233, 21523361, 43053282, 64570082, 129150085, 193710245, 387440172, 581130734, 1162291121
Offset: 0
- 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]
a(0)=1 prepended and terms a(36) and beyond from
Andrew Howroyd, Oct 01 2019
A056505
Number of periodic palindromic structures using a maximum of four different symbols.
Original entry on oeis.org
1, 1, 2, 2, 5, 5, 13, 15, 40, 51, 133, 187, 479, 715, 1815, 2795, 7036, 11051, 27747, 43947, 110061, 175275, 438683, 700075, 1750959, 2798251, 6997643, 11188907, 27975351, 44747435, 111876715, 178973355, 447445692, 715860651, 1789684267, 2863377067, 7158491875
Offset: 0
- 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]
a(0)=1 prepended and terms a(17) and beyond from
Andrew Howroyd, Oct 01 2019
A056506
Number of periodic palindromic structures using a maximum of five different symbols.
Original entry on oeis.org
1, 1, 2, 2, 5, 5, 13, 15, 41, 52, 143, 202, 564, 855, 2433, 3845, 11101, 18002, 52662, 86472, 255128, 422005, 1252898, 2079475, 6197590, 10306752, 30796243, 51263942, 153411664, 255514355, 765393943, 1275163905, 3822001141, 6368612302, 19095317552, 31821472612
Offset: 0
- 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]
a(0)=1 prepended and terms a(17) and beyond from
Andrew Howroyd, Oct 01 2019
A056507
Number of periodic palindromic structures using a maximum of six different symbols.
Original entry on oeis.org
1, 1, 2, 2, 5, 5, 13, 15, 41, 52, 144, 203, 577, 876, 2587, 4111, 12589, 20648, 65519, 109299, 356613, 601492, 2005545, 3403127, 11522829, 19628064, 67183531, 114700315, 395497312, 676207628, 2343265049, 4010090463, 13942486653, 23874362200, 83192239491, 142508723651
Offset: 0
- 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]
a(0)=1 prepended and terms a(16) and beyond from
Andrew Howroyd, Oct 01 2019
A056509
Number of periodic palindromic structures of length n using exactly three different symbols.
Original entry on oeis.org
0, 0, 0, 1, 1, 5, 6, 19, 25, 64, 90, 208, 301, 656, 966, 2035, 3025, 6250, 9330, 19035, 28501, 57740, 86526, 174436, 261625, 525994, 788970, 1583119, 2375101, 4760516, 7141686, 14303011, 21457825, 42954850, 64439010, 128953341, 193448101, 387046700, 580606446, 1161504423
Offset: 1
For example, aaabbb is not a (finite) palindrome but it is a periodic palindrome. 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]
A056508
Number of periodic palindromic structures of length n using exactly two different symbols.
Original entry on oeis.org
0, 1, 1, 3, 3, 6, 7, 13, 15, 25, 31, 50, 63, 99, 127, 197, 255, 391, 511, 777, 1023, 1551, 2047, 3090, 4095, 6175, 8191, 12323, 16383, 24639, 32767, 49221, 65535, 98431, 131071, 196743, 262143, 393471, 524287, 786697, 1048575, 1573375, 2097151, 3146255, 4194303
Offset: 1
From _Andrew Howroyd_, Apr 07 2017: (Start)
Example for n=6:
Periodic symmetry means results are either in the form abccba or abcdcb.
There are 3 binary words in the form abccba that start with 0 and contain a 1 which are 001100, 010010, 011110. Of these, 011110 is equivalent to 001100 after rotation.
There are 7 binary words in the form abcdcb that start with 0 and contain a 1 which are 000100, 001010, 001110, 010001, 010101, 011011, 011111. Of these, 011111 is equivalent to 000100, 010001 is equivalent to 001010 and 011011 is equivalent to 010010 from the first set.
There are therefore a total of 7 + 3 - 4 = 6 equivalence classes so a(6) = 6.
(End)
- 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]
-
(* b = A164090, c = A045674 *)
b[n_] := (1/4)*(7 - (-1)^n)*2^((1/4)*(2*n + (-1)^n - 1));
c[0] = 1;
c[n_] := c[n] = If[EvenQ[n], 2^(n/2 - 1) + c[n/2], 2^((n - 1)/2)];
a[n_] := If[OddQ[n], b[n]/2, (1/2)*(b[n] + c[n/2])] - 1;
Array[a, 45] (* Jean-François Alcover, Jun 29 2018, after Andrew Howroyd *)
A056510
Number of periodic palindromic structures of length n using exactly four different symbols.
Original entry on oeis.org
0, 0, 0, 0, 0, 1, 1, 7, 10, 43, 65, 220, 350, 1059, 1701, 4803, 7770, 21105, 34105, 90248, 145750, 379391, 611501, 1573432, 2532530, 6465473, 10391745, 26379908, 42355950, 107091559, 171798901, 433093459, 694337290, 1746630985, 2798806985, 7029341790, 11259666950
Offset: 1
For example, aaabbb is not a (finite) palindrome but it is a periodic palindrome. 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]
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