A177258
Number of derangements of {1,2,...,n} having no adjacent transpositions.
Original entry on oeis.org
1, 0, 0, 2, 7, 36, 225, 1610, 13104, 119548, 1208583, 13413960, 162176105, 2121703324, 29866022640, 450112042926, 7231658709455, 123388310103660, 2228221240575337, 42459591881035062, 851420058861276576, 17922280827967843160, 395141598274153826095
Offset: 0
a(4)=7 because we have (1342), (13)(24), (1324), (1432), (1423), (1234), and (1243).
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F:=Factorial;
A177258:= func< n | (&+[(&+[(-1)^(j+k)*F(n-k)/(F(j)*F(k)): k in [0..Floor((n-j)/2)]]): j in [0..n]]) >;
[A177258(n): n in [0..40]]; // G. C. Greubel, May 13 2024
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p := 1: q := 2: a := proc (n) local ct, t, s; ct := 0: for s from 0 to n/p do for t from 0 to n/q do if p*s+q*t <= n then ct := ct+(-1)^(s+t)*factorial(n-(p-1)*s-(q-1)*t)/(factorial(s)*factorial(t)) else end if end do end do: ct end proc; seq(a(n), n = 0 .. 22);
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p = 1; q = 2; a[n_] := Module[{ct, t, s}, ct = 0; For[s = 0, s <= n/p, s++, For[t = 0, t <= n/q, t++, If[p*s + q*t <= n, ct = ct + (-1)^(s+t) * Factorial[n - (p-1)*s - (q-1)*t]/(Factorial[s]*Factorial[t])]]]; ct];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Nov 24 2017, translated from Maple *)
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f=factorial;
def A177258(n): return sum(sum((-1)^(j+k)*f(n-k)/(f(j)*f(k)) for k in range(1+(n-j)//2)) for j in range(n+1))
[A177258(n) for n in range(41)] # G. C. Greubel, May 13 2024
A177259
Number of derangements of {1,2,...,n} having no adjacent 3-cycles (an adjacent 3-cycle is a cycle of the form (i,i+1,i+2)).
Original entry on oeis.org
1, 0, 1, 1, 9, 41, 258, 1809, 14575, 131660, 1320264, 14551987, 174887262, 2276174790, 31895551245, 478783042890, 7665081036273, 130370168718467, 2347620603019159, 44620121619435141, 892663172726141844, 18750621868455013979, 412602921349249182309
Offset: 0
a(5)=41 because among the 44 (= A000166(5)) derangements of {1,2,3,4,5} only (12)(345), (123)(45), and (15)(234) have adjacent 3-cycles.
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F:=Factorial;
A177258:= func< n | (&+[(&+[(-1)^(j+k)*F(n-2*k)/(F(j)*F(k)): k in [0..Floor((n-j)/3)]]): j in [0..n]]) >;
[A177258(n): n in [0..40]]; // G. C. Greubel, May 13 2024
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a := proc (n) local ct, t, s: ct := 0: for s from 0 to n do for t from 0 to (1/3)*n do if s+3*t <= n then ct := ct+(-1)^(s+t)*factorial(n-2*t)/(factorial(s)*factorial(t)) else end if end do end do: ct end proc; seq(a(n), n = 0 .. 22);
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a[n_] := Module[{ct = 0, t, s}, For[s = 0, s <= n, s++, For[t = 0, t <= n/3, t++, If[s + 3*t <= n, ct = ct + (-1)^(s + t)*Factorial[n - 2*t] / (Factorial[s]*Factorial[t])]]]; ct];
Table[a[n], {n, 0, 22}] (* Jean-François Alcover, Nov 24 2017, translated from Maple *)
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f=factorial;
def A177259(n): return sum(sum((-1)^(j+k)*f(n-2*k)/(f(j)*f(k)) for k in range(1+(n-j)//3)) for j in range(n+1))
[A177259(n) for n in range(41)] # G. C. Greubel, May 13 2024
A177261
Number of derangements of {1,2,...,n} having no adjacent 2-cycles and no adjacent 3-cycles (an adjacent q-cycle is a cycle of the form (i,i+1,i+2,...,i+q-1)).
Original entry on oeis.org
1, 0, 0, 1, 7, 35, 218, 1574, 12883, 117956, 1195590, 13295211, 160974037, 2108348871, 29704448652, 447997026724, 7201873573981, 122939256681704, 2221004487898100, 42336428273893565, 849195448479132811, 17879882855311478795, 394291291121879453430
Offset: 0
a(5)=35 because among the 44 (= A000166(5)) derangements of {1,2,3,4,5} only (12)(345), (12)(354), (145)(23), (154)(23), (125)(34), (152)(34), (123)(45), (132)(45) , and (15)(234) have adjacent 2-cycles or adjacent 3-cycles (or both).
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m:=30;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( (&+[Factorial(k)*(x*(1-x)/(1-x^4))^(k+1)/x: k in [0..m+2]]) )); // G. C. Greubel, May 13 2024
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a := proc (n) local ct, t, s, u: ct := 0: for s from 0 to n do for t from 0 to n do for u from 0 to n do if s+2*t+3*u <= n then ct := ct+(-1)^(s+t+u)*factorial(n-t-2*u)/(factorial(s)*factorial(t)*factorial(u)) else end if end do end do end do: ct end proc; seq(a(n), n = 0 .. 22);
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a[n_] := Sum[If[s + 2*t + 3*u <= n, (-1)^(s + t + u)*(n - t - 2 u)!/(s! t! u!), 0], {s, 0, n}, {t, 0, n}, {u, 0, n}];
Table[a[n], {n, 0, 22}] (* Jean-François Alcover, Dec 04 2017 *)
With[{m=40}, CoefficientList[Series[Sum[k!*(x*(1-x)/(1-x^4))^(k+1)/x,{k,0,m+2}], {x,0,m}], x]] (* G. C. Greubel, May 13 2024 *)
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m=30
def A177261_list(prec):
P. = PowerSeriesRing(ZZ, prec)
return P( sum(factorial(k)*(x*(1-x)/(1-x^4))^(k+1)/x for k in range(m+3)) ).list()
A177261_list(m) # G. C. Greubel, May 13 2024
Showing 1-3 of 3 results.