A190826 -id:A190826 - cp's OEIS frontend

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.

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

Seiichi Manyama, Nov 24 2018

			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.

Columns k=2..10 give A000012, A190917, A190918, A190920, A190923, A190927, A190932, A321987, A322061.
Rows n=1..10 give A000012, A278990, A190826, A190830, A190833, A190835, A190836, A190837, A321669, A321670.
Main diagonal gives A321666.
Cf. A322093, A369923.

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

T(n,k) = A322093(n,k) / k!. - Andrew Howroyd, Feb 03 2024

A190830 Number of permutations of 4 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, 182, 94376, 98371884, 182502973885, 551248360550999, 2536823683737613858, 16904301142107043464659, 156690501089429126239232946, 1955972150994131850032960933480, 32016987304767134806200915633253966, 672058204939482014438623912695190927357

Offset: 0

R. H. Hardin, May 21 2011

nonn

			Some solutions for n=3:
  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1  1
  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2  2
  3  1  3  3  1  3  3  3  3  3  3  1  1  3  1  1
  1  2  2  2  2  2  2  1  1  2  1  3  3  1  3  2
  2  3  3  3  1  1  3  2  2  3  2  1  2  3  1  1
  3  2  1  2  3  2  1  3  1  1  3  2  3  2  2  3
  1  3  2  3  2  3  3  1  3  2  2  3  2  1  3  1
  3  2  3  1  3  1  2  3  1  1  1  2  1  3  2  3
  1  1  1  3  1  3  1  2  2  3  3  3  2  2  3  2
  2  3  2  1  3  2  3  1  3  1  1  2  3  3  1  3
  3  1  1  2  2  3  1  3  2  2  2  3  1  1  3  2
  2  3  3  1  3  1  2  2  3  3  3  1  3  2  2  3

Seiichi Manyama, Table of n, a(n) for n = 0..158 (terms 1..28 from R. J. Mathar)
R. J. Mathar, A class of multinomial permutations avoiding object clusters, vixra:1511.0015 (2015), sequence M_{c,4}/c!.

Row n=4 of A322013.

Cf. A190826, A321633.

From Vaclav Kotesovec, Nov 24 2018: (Start)
Recurrence: 3*(64*n^3 - 280*n^2 + 414*n - 245)*a(n) = (2048*n^6 - 12032*n^5 + 30400*n^4 - 42608*n^3 + 32484*n^2 - 14624*n + 1731)*a(n-1) + 3*(3840*n^5 - 20640*n^4 + 40104*n^3 - 36340*n^2 + 23378*n - 13429)*a(n-2) - 18*(384*n^4 - 1488*n^3 + 1556*n^2 - 986*n + 649)*a(n-3) - 27*(64*n^3 - 88*n^2 + 46*n - 47)*a(n-4).
a(n) ~ 2^(5*n+1) * n^(3*n) / (3^n * exp(3*n + 3)). (End)

a(0)=1 prepended by Seiichi Manyama, Nov 16 2018

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

Michael De Vlieger, Nov 01 2021

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..200
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. See Appendix Table 2.

Row 3 of A369923.

Cf. A190826, A348814, A292409; A003436, A348815, A348818, A348821.

a(14) onwards from Andrew Howroyd, Feb 05 2024

A193624 Number of ways n triples can sit in a row without any siblings next to each other.

Original entry on oeis.org

1, 0, 72, 37584, 53529984, 152458744320, 766958183193600, 6236531290739312640, 76788695692068062330880, 1361934174627779827740180480, 33454092372947487842682293452800, 1102556254139040688616563751190528000

Offset: 0

Andrew Woods, Aug 01 2011

nonn

Andrew Woods, Table of n, a(n) for n = 0..101

Cf. A007060, A190826, A330266.

B:=Binomial;
f:= func< n,j | (&+[B(n,k)*B(2*k,j)*(-3)^(n+k-j): k in [Ceiling(j/2)..n]]) >;
A193624:= func< n | (&+[Factorial(n+j)*f(n,j): j in [0..2*n]]) >;
[A193624(n): n in [0..30]]; // G. C. Greubel, Sep 22 2023

A193624[n_]:= Sum[(n+j)!*Binomial[n,k]*Binomial[2*k,j]*(-3)^(n+k-j), {j,0,2*n}, {k,Ceiling[j/2],n}];
Array[A193624, 30, 0] (* G. C. Greubel, Sep 22 2023 *)

b=binomial;
def f(j,n): return sum(b(n,k)*b(2*k,j)*(-3)^(n+k-j) for k in range((j//2),n+1))
def A193624(n): return sum(factorial(n+j)*f(j,n) for j in range(2*n+1))
[A193624(n) for n in range(31)] # G. C. Greubel, Sep 22 2023

Lim_{n -> oo} a(n) -> (3n)!*exp(-2).
a(n) = A190826(n) * 6^n * n! for n >= 1. - Nathaniel Johnston, Aug 01 2011
a(n) -3*(9*n^3-9*n^2+8*n+8)*a(n-1) +108*(n-1)*(n^2-11*n+16)*a(n-2) +3024*(n-1)*(n-2)^2*a(n-3) -5184*(n-1)*(n-2)*(n-3)*a(n-4) = 0. - R. J. Mathar, May 23 2014
a(n) = Sum_{j=0..2*n} Sum_{k=ceiling(j/2)..n} (n+j)! * binomial(2*k, j) * binomial(n, k) * (-3)^(n+k-j). - G. C. Greubel, Sep 22 2023

A193638 Number of permutations of the multiset {1,1,1,2,2,2,3,3,3,...,n,n,n} with no two consecutive terms equal.

Original entry on oeis.org

1, 0, 2, 174, 41304, 19606320, 16438575600, 22278418248240, 45718006789687680, 135143407245840698880, 553269523327347306412800, 3039044104423605600086688000, 21819823367694505460651694873600, 200345011881335747639978525387827200

Offset: 0

Andrew Woods, Aug 01 2011

nonn

			a(2) = 2 because there are two permutations of {1,1,1,2,2,2} avoiding equal consecutive terms: 121212 and 212121.

Andrew Woods, Table of n, a(n) for n = 0..101
H. Eriksson and A. Martin, Enumeration of Carlitz multipermutations, arXiv:1702.04177 [math.CO], 2017.

Cf. A114938 = similar, with two copies instead of three.
Cf. A193624 = arrangements of triples with no adjacent siblings.
Cf. A190826.

B:=Binomial;
f:= func< n,j | (&+[B(n,k)*B(2*k,j)*(-3)^(k-j): k in [Ceiling(j/2)..n]]) >;
A193638:= func< n | (-1/2)^n*(&+[Factorial(n+j)*f(n,j): j in [0..2*n]]) >;
[A193638(n): n in [0..30]]; // G. C. Greubel, Sep 22 2023

a:= proc(n) option remember; `if`(n<3, (n-1)*(3*n-2)/2,
    n*((3*n-1)*(3*n^2-5*n+4) *a(n-1) +2*(n-1)*(6*n^2-9*n-1) *a(n-2)
    -4*n*(n-1)*(n-2) *a(n-3))/(2*n-2))
    end:
seq(a(n), n=0..20);  # Alois P. Heinz, Jun 05 2013

a[n_]:= (1/6^n)*Sum[(n+j)!*Binomial[n, k]*Binomial[2k, j]*(-3)^(n+k-j), {j,0,2n}, {k, Ceiling[j/2], n}]; Table[a[n], {n, 0, 13}] (* Jean-François Alcover, Jul 22 2017, after Tani Akinari *)

a(n):= (1/6^n)*sum((n+j)!*sum(binomial(n,k)*binomial(2*k,j)* (-3)^(n+k-j), k,ceiling(j/2),n), j,0,2*n); /* Tani Akinari, Sep 23 2012 */

from sympy.core.cache import cacheit
@cacheit
def a(n): return (n-1)*(3*n-2)//2 if n<3 else n*((3*n-1)*(3*n**2 - 5*n + 4)*a(n-1) + 2*(n-1)*(6*n**2 -9*n-1)*a(n-2) - 4*n*(n-1)*(n-2)*a(n- 3))//(2*n-2)
print([a(n) for n in range(51)]) # Indranil Ghosh, Jul 22 2017, formula after Maple code

b=binomial;
def f(j,n): return sum(b(n,k)*b(2*k,j)*(-3)^(k-j) for k in range((j//2),n+1))
def A193638(n): return (-1/2)^n*sum(factorial(n+j)*f(j,n) for j in range(2*n+1))
[A193638(n) for n in range(31)] # G. C. Greubel, Sep 22 2023

a(n) = A190826(n) * n! for n >= 1.
a(n) = A193624(n)/6^n.
a(n) = Sum_{s+t+u=n} (-1)^t*multinomial(n;s,t,u)*(3*s+2*t+u)!/(3!)^s. - Alexis Martin, Nov 16 2017
a(n) = (1/6^n) * Sum_{j=0..2*n} Sum_{k=ceiling(j/2)..n} (n+j)! * binomial(2*k, j) * binomial(n, k) * (-3)^(n+k-j). - Tani Akinari, Sep 23 2012
a(n) = n*( (3*n-1)*(3*n^2-5*n+4)*a(n-1) +2*(n-1)*(6*n^2-9*n-1)*a(n-2) -4*n*(n-1)*(n-2)*a(n-3) )/(2*n-2). - Alois P. Heinz, Jun 05 2013

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

Seiichi Manyama, Nov 16 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..27

Main diagonal of A322013.

Cf. A190826, A190830, A190833, A190835, A190836, A190837, A278990, A321634.

{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

a(n) = A321634(n)/n!.

a(n) ~ exp(5/12) * n^((n-1)*(2*n-1)/2) / (2*Pi)^(n/2). - Vaclav Kotesovec, Nov 24 2018

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

Seiichi Manyama, Nov 16 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..70

Row n=9 of A322013.

Cf. A190826, A190830, A190833, A190835, A190836, A190837, A321670.

a(n) ~ 9^(7*n + 1/2) * n^(8*n) / (4480^n * exp(8*(n+1))). - Vaclav Kotesovec, Nov 24 2018

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

Seiichi Manyama, Nov 16 2018

nonn

In general, for r > 1, row r of A322013 is asymptotic to r^(r*n + 1/2) * n^((r-1)*n) / ((r!)^n * exp((r-1)*(n+1))). - Vaclav Kotesovec, Nov 24 2018

Seiichi Manyama, Table of n, a(n) for n = 0..63

Cf. A190826, A190830, A190833, A190835, A190836, A190837, A321669.

Row r=10 of A322013.

a(n) ~ 2^(2*n + 1/2) * 5^(8*n + 1/2) * n^(9*n) / (567^n * exp(9*(n+1))). - Vaclav Kotesovec, Nov 24 2018

A348814 a(n) = number of loopless diagrams by number of K_3 up to rotational symmetry.

Original entry on oeis.org

0, 1, 4, 126, 9367, 1120780, 189565588, 43154533233, 12735808866899, 4732638168795171, 2163220895025390670, 1193176166690983987122, 781607533669746761791541

Offset: 1

Michael De Vlieger, Nov 01 2021

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. See Appendix Table 2.

Cf. A190826, A292409, A348813; A007474, A348816, A348819, A348822.

cp's OEIS Frontend

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.

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A190830 Number of permutations of 4 copies of 1..n introduced in order 1..n with no element equal to another within a distance of 1.

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A348813 a(n) = number of chord labeled loopless diagrams by number of K_3.

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A193624 Number of ways n triples can sit in a row without any siblings next to each other.

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A193638 Number of permutations of the multiset {1,1,1,2,2,2,3,3,3,...,n,n,n} with no two consecutive terms equal.

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A321666 Number of arrangements of n 1's, n 2's, ..., n n's avoiding equal consecutive terms and introduced in ascending order.

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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.

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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.

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A348814 a(n) = number of loopless diagrams by number of K_3 up to rotational symmetry.

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