A322013
Square array A(n,k), n >= 1, k >= 1, read by antidiagonals, where A(n,k) is the number of permutations of n copies of 1..k introduced in order 1..k with no element equal to another within a distance of 1.
Original entry on oeis.org
1, 1, 0, 1, 1, 0, 1, 5, 1, 0, 1, 36, 29, 1, 0, 1, 329, 1721, 182, 1, 0, 1, 3655, 163386, 94376, 1198, 1, 0, 1, 47844, 22831355, 98371884, 5609649, 8142, 1, 0, 1, 721315, 4420321081, 182502973885, 66218360625, 351574834, 56620, 1, 0
Offset: 1
Square array begins:
1, 1, 1, 1, 1, 1, ...
0, 1, 5, 36, 329, 3655, ...
0, 1, 29, 1721, 163386, 22831355, ...
0, 1, 182, 94376, 98371884, 182502973885, ...
0, 1, 1198, 5609649, 66218360625, 1681287695542855, ...
0, 1, 8142, 351574834, 47940557125969, 16985819072511102549, ...
- Seiichi Manyama, Antidiagonals n = 1..53, flattened
- Evgeniy Krasko, Igor Labutin, and Alexander Omelchenko, Enumeration of Labelled and Unlabelled Hamiltonian Cycles in Complete k-partite Graphs, arXiv:1709.03218 [math.CO], 2017.
- Mathematics.StackExchange, Find the number of k 1's, k 2's, ... , k n's - total kn cards, Apr 08 2012.
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q(n,x) = sum(i=1, n, (-1)^(n-i) * binomial(n-1, n-i) * x^i/i!)
T(n,k) = subst(serlaplace(q(n,x)^k), x, 1)/k! \\ Andrew Howroyd, Feb 03 2024
A348815
a(n) = number of chord labeled loopless diagrams by number of K_4.
Original entry on oeis.org
0, 1, 134, 75843, 83002866, 158861646466, 490294453324924, 2292204611710892971, 15459367618357013402267, 144663877588996810362218074, 1819753109993633276315632934129, 29976383544377113242613349012354566, 632574848906117234957565158900144038734
Offset: 1
A348813
a(n) = number of chord labeled loopless diagrams by number of K_3.
Original entry on oeis.org
0, 1, 22, 1415, 140343, 20167651, 3980871156, 1035707510307, 343866839138005, 141979144588872613, 71386289535825383386, 42954342000612934599071, 30482693813120122213093587, 25196997894058490607106028095, 24001522306527907199721466108488, 26102037346800387738363882455862531
Offset: 1
A348818
a(n) = number of chord labeled loopless diagrams by number of K_5.
Original entry on oeis.org
0, 1, 866, 4446741, 55279816356, 1450728060971387, 72078730629785795963, 6235048155225093080061949, 879601407931825739964190440635, 192100729970218737700046212217095291, 62258393664270652226502315136978421947948, 28913744296806659870889046765907226809528931041
Offset: 1
A348821
a(n) = number of chord labeled loopless diagrams by number of K_6.
Original entry on oeis.org
0, 1, 5812, 276154969, 39738077935264, 14571371516350429940, 11876790400066163254723167, 19372051918038657958659363247949, 58256941603805590330534264712744407687, 302616041649108508974263266688425815263488561, 2575195630881373033515248134269171034879932771154311
Offset: 1
A369925
Number of uniform circular words of length n with adjacent elements unequal using an infinite alphabet up to permutations of the alphabet.
Original entry on oeis.org
1, 0, 1, 1, 2, 1, 6, 1, 33, 23, 295, 1, 4877, 1, 44191, 141210, 749316, 1, 31762349, 1, 309754506, 3980911205, 4704612121, 1, 1303743206944, 55279816357, 2737023412201, 343866841144704, 564548508168226, 1, 145630899385513158, 1, 2359434158555273239
Offset: 0
a(1) = 0 because the symbol 'a' is considered to be adjacent to itself in a circular word. The set partition {{1}} is also excluded because 1 == 1 + 1 (mod 1).
The a(6) = 6 words are ababab, abacbc, abcabc, abcacb, abcbac, abcdef.
The corresponding a(6) = 6 set partitions are:
{{1,3,5},{2,4,6}},
{{1,3},{2,5},{4,6}},
{{1,4},{2,5},{3,6}},
{{1,4},{2,6},{3,5}},
{{1,5},{2,4},{3,6}},
{{1},{2},{3},{4},{5},{6}}.
The case for adjacent elements possibly equal is
A038041.
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\\ Needs T(n,k) from A369923.
a(n) = {if(n==0, 1, sumdiv(n, d, T(d, n/d)))}
A378241
Numbers of directed Hamiltonian cycles in the complete 4-partite graph K_{n,n,n,n}.
Original entry on oeis.org
6, 1488, 3667680, 37744330752, 1106491456512000, 74213488705904640000, 9872975878366503813120000, 2355966665497190945783808000000, 935825492908108988335792827924480000, 584053924678169568704863421815848960000000
Offset: 1
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Table[Sum[Sum[Sum[Sum[ (2n - i - j - 1)! 2^(2i) 3^j (n!)^4/(j!) * (3n - 3i - 3j - 2d)!/((2i + j - n + d)! (n - j - d)! (2n - 3i - 2j - d)!) * (2n - 2i - 2j - 2e)!/(e! (d - e)! (2n - 2i - 2j - d - e)! (n - i - j - d + e)! ((n - i - j - e)!)^2), {e, Max[0, i + j - n + d], Min[d, 2n - 2i - 2j - d]}], {d, Max[0, n - j - 2i], Min[n - j, 2n - 3i - 2j]}], {i, 0, Floor[2(n - j)/3]}], {j, 0, n}], {n, 1, 10}]
Table[(n!)^4 Expand[Hypergeometric1F1[1 - n, 2, x]^4 x^3] /. x^p_ :> p!, {n, 10}] (* Eric W. Weisstein, Feb 20 2025 *)
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from math import factorial as fact
def a(n):
# Using formula found in Horak et al.
return sum(sum(sum(sum(
fact(2*n-i-j-1)*pow(2,2*i)*pow(3,j)*pow(fact(n),4)//fact(j) *
fact(3*n-3*i-3*j-2*d)//(fact(2*i+j-n+d)*fact(n-j-d)*fact(2*n-3*i-2*j-d)) *
fact(2*n-2*i-2*j-2*e)//(fact(e)*fact(d-e)*fact(2*n-2*i-2*j-d-e)*fact(n-i-j-d+e)*pow(fact(n-i-j-e),2))
for e in range(max(0,i+j-n+d), min(d,2*n-2*i-2*j-d)+1))
for d in range(max(0,n-j-2*i), min(n-j,2*n-3*i-2*j)+1))
for i in range(int(2*(n-j)/3)+1))
for j in range(n+1))
print([a(n) for n in range(1,11)])
Showing 1-7 of 7 results.
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