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
A321633
Number of permutations of the multiset {1,1,1,1,2,2,2,2,3,3,3,3,...,n,n,n,n} with no two consecutive terms equal.
Original entry on oeis.org
1, 0, 2, 1092, 2265024, 11804626080, 131402141197200, 2778291737177034960, 102284730928300590754560, 6134232798447803932455457920, 568598490353320413296928514444800, 78076149156802562231395694989534464000, 15336188146163145199585928509793662920345600
Offset: 0
a(2) = 2 because there are two permutations of {1,1,1,1,2,2,2,2} avoiding equal consecutive terms: 12121212 and 21212121.
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a[n_] := Integrate[(-x + 3/2 * x^2 - 1/2 * x^3 + 1/24 * x^4)^n * Exp[-x], {x, 0, Infinity}]; Array[a, 10, 0] (* Stefano Spezia, Nov 27 2018 *)
A348816
a(n) = number of loopless diagrams by number of K_4 up to rotational symmetry.
Original entry on oeis.org
0, 1, 15, 4790, 4151415, 6619291247, 17510518983528, 71631394311300461, 429426878302882412435, 3616596939726424941979785
Offset: 1
A321666
Number of arrangements of n 1's, n 2's, ..., n n's avoiding equal consecutive terms and introduced in ascending order.
Original entry on oeis.org
1, 1, 1, 29, 94376, 66218360625, 16985819072511102549, 2421032324142610480402567434373, 271259741131895052775392614041761701799270286, 32119646666355552112999645991677870426882424139287301894021793
Offset: 0
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{a(n) = sum(i=n, n^2, i!*polcoef(sum(j=1, n, (-1)^(n-j)*binomial(n-1, j-1)*x^j/j!)^n, i))/n!} \\ Seiichi Manyama, May 27 2019
A321669
Number of permutations of 9 copies of 1..n introduced in order 1..n with no element equal to another within a distance of 1.
Original entry on oeis.org
1, 0, 1, 2872754, 104650147201049, 23575497690601916022516, 24302858067615766089801166488125, 91155245844064069307740171414201519055298, 1046031892354833895113128900608177633584652958677057, 32119646666355552112999645991677870426882424139287301894021793
Offset: 0
A321670
Number of permutations of 10 copies of 1..n introduced in order 1..n with no element equal to another within a distance of 1.
Original entry on oeis.org
1, 0, 1, 20824778, 7279277647839552, 19672658572012343899666292, 293736218147318801678882792470437721, 18739368045280595665934917472507368174737872589, 4204427313459831775866154680419213479057724331798640498651
Offset: 0
A348817
a(n) = number of loopless diagrams by number of K_4 up to all symmetries.
Original entry on oeis.org
0, 1, 13, 2576, 2081393, 3309962320, 8755277273334, 35815698613833466, 214713439275724149414, 1808298469877117320495867
Offset: 1
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