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A256033
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Number of idempotents of rank 1 in partition monoid P_n.
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4
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1, 5, 43, 529, 8451, 167397, 3984807, 111319257, 3583777723, 131082199809, 5385265586075, 246172834737485, 12422776100542887, 687441750763500441, 41475644663003037947, 2714680813135603845921
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OFFSET
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1,2
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LINKS
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MAPLE
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e256033 := proc(n, r, s)
option remember;
local resu, m, a, b;
if n <= 0 then
return 0;
end if;
if s = 1 then
combinat[stirling2](n, r) ;
elif r= 1 then
combinat[stirling2](n, s) ;
else
resu := s*procname(n-1, r-1, s)+r*procname(n-1, r, s-1)+r*s*procname(n-1, r, s) ;
for m from 1 to n-2 do
for a from 1 to r-1 do
for b from 1 to s-1 do
resu := resu + binomial(n-2, m) *(a*(s-b)+b*(r-a))
*procname(m, a, b)*procname(n-m-1, r-a, s-b);
end do:
end do:
end do:
resu ;
end if;
end proc:
a := 0 ;
for r from 1 to n do
for s from 1 to n do
a := a+r*s*e256033(n, r, s) ;
end do;
end do;
end proc:
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MATHEMATICA
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f[n_, r_, s_] := f[n, r, s] = Module[{resu, m, a, b}, Which[n <= 0, 0, s == 1, StirlingS2[n, r], r == 1, StirlingS2[n, s], True, resu = s*f[n - 1, r - 1, s] + r*f[n - 1, r, s - 1] + r*s*f[n - 1, r, s]; Do[resu += Binomial[n - 2, m]*(b*(r - a) + a*(s - b))*f[m, a, b]*f[-m + n - 1, r - a, s - b], {m, n}, {a, r - 1}, {b, s - 1}]; resu]];
a[n_] := Module[{b = 0}, Do[b += r*s*f[n, r, s], {r, n}, {s, n}]; b];
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PROG
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(Sage)
@cached_function
def F(n, r, s):
if n <= 0: return 0
if s == 1: return stirling_number2(n, r)
if r == 1: return stirling_number2(n, s)
ret = s*F(n-1, r-1, s)+r*F(n-1, r, s-1)+r*s*F(n-1, r, s)
for m in (1..n-2):
for a in (1..r-1):
for b in (1..s-1):
ret += binomial(n-2, m)*(a*(s-b)+b*(r-a))*F(m, a, b)*F(n-m-1, r-a, s-b)
return ret
@cached_function
a = 0
for r in (1..n):
for s in (1..n):
a += r*s*F(n, r, s)
return a
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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