A190918 -id:A190918 - 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

A190917 Number of permutations of n copies of 1..3 introduced in order 1..3 with no element equal to another within a distance of 1.

Original entry on oeis.org

1, 1, 5, 29, 182, 1198, 8142, 56620, 400598, 2872754, 20824778, 152303410, 1122149800, 8319825040, 62017475600, 464452683432, 3492568119566, 26358270711370, 199565061455634, 1515311001158482, 11535716330003876, 88025068713285476, 673124069796140900

Offset: 0

R. H. Hardin, May 23 2011

nonn

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

Seiichi Manyama, Table of n, a(n) for n = 0..1111 (terms 1..200 from R. H. Hardin)
R. J. Mathar, A class of multinomial permutations avoiding object clusters, vixra:1511.0015 (2015), sequence M_{3,m}/3!.

Column 3 of A322013.

Cf. A000012 (b=2), A190918 (b=4), A190920 (b=5), A190923 (b=6), A190927 (b=7), A190932 (b=8), A321987 (b=9), A322061 (b=10).

[(&+[Binomial(n-1, k)*(Binomial(n-1, k)*Binomial(2*n+1-2*k, n+1) + Binomial(n-1, k+1)*Binomial(2*n-2*k, n+1)): k in [0..Floor(n/2)]])/3: n in [1..25]]; // G. C. Greubel, Nov 24 2018

a:= proc(n) option remember; `if`(n<3, [1$2, 5][n+1],
      ((7*n-4)*a(n-1)+8*(n-2)^2*a(n-2)/(n+1))/n)
    end:
seq(a(n), n=0..22);  # Alois P. Heinz, Sep 09 2023

Table[(1/3)*Sum[Binomial[n-1, k]*(Binomial[n-1, k]*Binomial[2*n+1-2*k, n+1] + Binomial[n-1, k+1]*Binomial[2*n-2*k, n+1]), {k,0,Floor[n/2]}], {n,1,25}] (* G. C. Greubel, Nov 24 2018 *)

A190917(n) = sum(k=0, n\2, binomial(n-1, k)*(binomial(n-1, k)*binomial(2*n+1-2*k, n+1)+binomial(n-1, k+1)*binomial(2*n-2*k, n+1))) / 3; \\ Max Alekseyev, Dec 10 2017

[(1/3)*sum(binomial(n-1, k)*(binomial(n-1, k)*binomial(2*n+1-2*k, n+1) + binomial(n-1, k+1)*binomial(2*n-2*k, n+1))  for k in range(1+floor(n/2))) for n in (1..25)] # G. C. Greubel, Nov 24 2018

a(n) = A110706(n) / 6 for n >= 1.

n*(n+1)*a(n) - (n+1)*(7*n-4)*a(n-1) - 8*(n-2)^2*a(n-2) = 0. - R. J. Mathar, Nov 01 2015 from A110706

a(0)=1 prepended by Alois P. Heinz, Sep 09 2023

A190920 Number of permutations of n copies of 1..5 introduced in order 1..5 with no element equal to another within a distance of 1.

Original entry on oeis.org

1, 329, 163386, 98371884, 66218360625, 47940557125969, 36533294879349056, 28920026907938624194, 23575497690601916022516, 19672658572012343899666292, 16730974132035148942028759656, 14455459908454408519322566567054

Offset: 1

R. H. Hardin, May 23 2011

nonn

			Some solutions for n=2
..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....3....3....3....3....3....3....3....1....3....3....3....3....3....3....3
..4....4....2....1....4....4....4....4....3....4....4....4....1....4....4....4
..1....5....4....4....1....5....3....5....4....5....3....5....4....3....2....5
..5....3....5....2....5....2....4....4....5....2....2....3....5....2....3....2
..2....2....4....3....4....1....5....5....2....4....5....5....4....5....5....5
..5....1....3....5....2....3....1....1....4....3....1....4....2....1....4....4
..3....4....5....4....3....4....5....3....5....5....5....1....5....4....1....3
..4....5....1....5....5....5....2....2....3....1....4....2....3....5....5....1

Seiichi Manyama, Table of n, a(n) for n = 1..334 (terms 1..55 from R. H. Hardin)

Column k=5 of A322013.

Cf. A000012 (b=2), A190917 (b=3), A190918 (b=4), A190923 (b=6), A190927 (b=7), A190932 (b=8), A321987 (b=9).

a(n) ~ 25 * sqrt(5) * 2^(10*n-7) / (27 * Pi^2 * n^2). - Vaclav Kotesovec, Nov 24 2018

A190923 Number of permutations of n copies of 1..6 introduced in order 1..6 with no element equal to another within a distance of 1.

Original entry on oeis.org

1, 3655, 22831355, 182502973885, 1681287695542855, 16985819072511102549, 183095824753841610373405, 2070756746775910218326948065, 24302858067615766089801166488125, 293736218147318801678882792470437721

Offset: 1

R. H. Hardin, May 23 2011

nonn

			Some solutions for n=2
..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....3....3....3....3....3....3....3....3....3....3....3....3....3....3....3
..4....2....4....4....4....2....4....4....4....4....4....4....4....4....4....4
..5....4....5....5....5....4....1....5....5....2....5....1....5....5....1....3
..1....5....4....3....3....1....3....4....2....5....6....4....6....6....5....4
..4....6....2....2....6....4....5....5....4....6....1....5....3....2....2....2
..5....3....1....6....5....5....6....1....5....5....5....6....2....6....6....5
..2....5....6....5....6....3....5....6....3....3....3....2....6....3....3....1
..6....6....5....6....2....6....4....3....6....4....6....5....4....1....6....6
..3....1....6....1....1....5....2....6....1....1....2....6....5....4....5....5
..6....4....3....4....4....6....6....2....6....6....4....3....1....5....4....6

Seiichi Manyama, Table of n, a(n) for n = 1..240 (terms 1..35 from R. H. Hardin)

Column k=6 of A322013.

Cf. A000012 (b=2), A190917 (b=3), A190918 (b=4), A190920 (b=5), A190927 (b=7), A190932 (b=8), A321987 (b=9).

a(n) ~ 9 * 5^(6*n-2) / (128 * sqrt(2) * Pi^(5/2) * n^(5/2)). - Vaclav Kotesovec, Nov 24 2018

A190927 Number of permutations of n copies of 1..7 introduced in order 1..7 with no element equal to another within a distance of 1.

Original entry on oeis.org

1, 47844, 4420321081, 551248360550999, 81644850343968535401, 13519747358522016160671387, 2421032324142610480402567434373, 459408385876250801291447710561829082, 91155245844064069307740171414201519055298

Offset: 1

R. H. Hardin, May 23 2011

nonn

			Some solutions for n=2:
  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  3  3  3  3  3  3  3  3  3  3  3  3  3  3  3
  4  4  4  4  4  2  2  4  4  4  2  4  4  4  4  2
  5  5  5  1  5  4  4  2  5  5  4  2  1  3  2  1
  6  6  6  5  6  5  3  5  3  6  5  4  4  5  5  4
  2  7  4  6  7  6  5  6  6  3  6  5  5  6  6  5
  7  3  5  4  2  3  6  5  7  7  7  1  6  4  4  6
  5  4  2  2  1  5  7  6  2  6  3  6  7  5  7  7
  6  5  6  3  6  4  5  1  4  1  7  3  2  7  1  3
  4  6  1  5  7  7  7  3  7  7  5  7  5  2  7  4
  3  2  7  7  5  1  1  7  1  5  1  6  7  1  5  6
  7  7  3  6  4  6  6  4  5  4  6  5  6  7  6  7
  1  1  7  7  3  7  4  7  6  2  4  7  3  6  3  5

Seiichi Manyama, Table of n, a(n) for n = 1..185 (terms 1..25 from R. H. Hardin)

Column k=7 of A322013.

Cf. A000012 (b=2), A190917 (b=3), A190918 (b=4), A190920 (b=5), A190923 (b=6), A190932 (b=8), A321987 (b=9).

a(n) ~ 343 * sqrt(7) * 2^(7*n-8) * 3^(7*n-3) / (625 * Pi^3 * n^3). - Vaclav Kotesovec, Nov 24 2018

A190932 Number of permutations of n copies of 1..8 introduced in order 1..8 with no element equal to another within a distance of 1.

Original entry on oeis.org

1, 721315, 1133879136649, 2536823683737613858, 6945222145021508480249929, 21671513613423101256198918372909, 74115215422015289392187745053216373265, 271259741131895052775392614041761701799270286

Offset: 1

R. H. Hardin, May 23 2011

nonn

			Some solutions for n=2
..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
..1....3....3....1....1....1....3....1....3....3....1....1....1....1....3....1
..3....1....1....3....3....3....1....3....1....1....3....3....3....3....1....2
..4....4....4....4....4....4....4....4....2....4....4....4....4....4....4....3
..5....5....5....3....5....5....5....5....3....5....5....5....2....5....5....4
..6....4....3....5....6....6....6....4....4....3....4....6....5....6....2....5
..5....6....6....6....7....7....2....6....5....6....5....7....6....5....6....6
..6....7....7....7....5....2....3....7....6....5....6....8....7....7....7....7
..3....8....8....8....6....8....5....8....7....2....7....7....5....8....3....8
..2....5....2....7....8....6....7....3....4....7....2....5....8....3....8....3
..7....2....6....8....2....3....6....6....6....6....6....2....3....6....4....5
..8....8....7....4....3....4....8....5....8....7....7....3....7....4....6....7
..4....3....4....6....4....5....7....8....5....8....8....4....4....2....7....4
..8....7....5....5....8....8....8....2....7....4....3....8....8....7....8....8
..7....6....8....2....7....7....4....7....8....8....8....6....6....8....5....6

Seiichi Manyama, Table of n, a(n) for n = 1..149 (terms 1..20 from R. H. Hardin)

Column k=8 of A322013.

Cf. A000012, A190917, A190918, A190920, A190923, A190927, A321987.

a(n) ~ 2 * 7^(8*n-2) / (1215 * sqrt(3) * Pi^(7/2) * n^(7/2)). - Vaclav Kotesovec, Nov 24 2018

A321987 Number of permutations of n copies of 1..9 introduced in order 1..9 with no element equal to another within a distance of 1.

Original entry on oeis.org

1, 12310199, 372419001449076, 16904301142107043464659, 967335448974819561548523580438, 64311863997340571475504065539218471107, 4749303210651587675797285013227098386984170468, 379065045836307787068046364731543393514652159389593652

Offset: 1

Seiichi Manyama, Nov 23 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..100

Column k=9 of A322013.

Cf. A000012, A190917, A190918, A190920, A190923, A190927, A190932.

a(n) ~ 2187 * 2^(27*n-14) / (84035 * Pi^4 * n^4). - Vaclav Kotesovec, Nov 24 2018

A377586 Numbers of directed Hamiltonian paths in the complete 4-partite graph K_{n,n,n,n}.

Original entry on oeis.org

24, 13824, 53529984, 751480602624, 27917203599360000, 2267561150913576960000, 354252505303682314076160000, 97087054992658680467800719360000, 43551509948777170973522371396239360000, 30293653795894300342540281328749772800000000

Offset: 1

Zlatko Damijanic, Nov 02 2024

nonn

Zlatko Damijanic, Table of n, a(n) for n = 1..117
L. Q. Eifler, K. B. Reid Jr., and D. P. Roselle, Sequences with adjacent elements unequal, Aequationes Mathematicae 6 (2-3), 1971; see also.

Cf. A190918, A234365, A322013, A322093.

Table[n!^4 * SeriesCoefficient[1/(1 - Sum[x[i]/(1 + x[i]), {i, 1, 4}]), Sequence @@ Table[{x[i], 0, n}, {i, 1, 4}]], {n, 1, 10}]

from math import factorial as fact, comb
from itertools import combinations_with_replacement
def a(n):
    #  Using modified formula for counting sequences found in Eifler et al.
    result = 0
    fn = fact(n)
    for i, j, k in combinations_with_replacement(range(1, n+1), 3):
        patterns = [(3,0,0)] if i == j == k else \
          [(2,0,1)] if i == j != k else \
          [(1,2,0)] if i != j == k else [(1,1,1)]
        for a, b, c in patterns:
            s = a*i + b*j + c*k
            num = fact(3)
            den = fact(a) * fact(b) * fact(c)
            if a:
                for _ in range(a): num, den = num * comb(n-1, i-1), den * fact(i)
            if b:
                for _ in range(b): num, den = num * comb(n-1, j-1), den * fact(j)
            if c:
                for _ in range(c): num, den = num * comb(n-1, k-1), den * fact(k)
            num *= comb(s + 1, n) * fact(s)
            result += (1 if (3*n - s) % 2 == 0 else -1) * (num // den)
    for _ in range(4): result *= fn
    return result
print([a(n) for n in range(1,11)]) # Zlatko Damijanic, Nov 18 2024

a(n) = 24 * n!^4 * A190918(n).

a(n) = n!^4 * A322093(n,4).

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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A190917 Number of permutations of n copies of 1..3 introduced in order 1..3 with no element equal to another within a distance of 1.

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A190920 Number of permutations of n copies of 1..5 introduced in order 1..5 with no element equal to another within a distance of 1.

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A190923 Number of permutations of n copies of 1..6 introduced in order 1..6 with no element equal to another within a distance of 1.

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A190927 Number of permutations of n copies of 1..7 introduced in order 1..7 with no element equal to another within a distance of 1.

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A190932 Number of permutations of n copies of 1..8 introduced in order 1..8 with no element equal to another within a distance of 1.

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A321987 Number of permutations of n copies of 1..9 introduced in order 1..9 with no element equal to another within a distance of 1.

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A377586 Numbers of directed Hamiltonian paths in the complete 4-partite graph K_{n,n,n,n}.

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