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A240797
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Total number of occurrences of the pattern 1=2 in all preferential arrangements (or ordered partitions) of n elements.
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2
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0, 1, 9, 78, 750, 8115, 98343, 1324204, 19650060, 318926745, 5623615965, 107093749818, 2191142272410, 47944109702671, 1117341011896515, 27633982917342360, 722929036749464280, 19946727355457792853, 578926427416920550233, 17632301590672398115270
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OFFSET
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1,3
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COMMENTS
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The number that avoid the pattern 1=2 is n! (these are the permutations on n elements).
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LINKS
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FORMULA
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EXAMPLE
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The 13 preferential arrangements on 3 points and the number of times the pattern 1=2 occurs are:
1<2<3, 0
1<3<2, 0
2<1<3, 0
2<3<1, 0
3<1<2, 0
3<2<1, 0
1=2<3, 1
1=3<2, 1
2=3<1, 1
1<2=3, 1
2<1=3, 1
3<1=2, 1
1=2=3, 3,
for a total of a(3) = 9.
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MAPLE
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b:= proc(n) option remember; `if`(n=0, [1, 0], add((p-> p+
[0, p[1]*j*(j-1)/2])(b(n-j))*binomial(n, j), j=1..n))
end:
a:= n-> b(n)[2]:
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MATHEMATICA
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b[n_] := b[n] = If[n == 0, {1, 0}, Sum[Function[p, p + {0, p[[1]]*j*(j - 1)/2}][b[n - j]]*Binomial[n, j], {j, 1, n}]]; a[n_] := b[n][[2]]; Table[a[n], {n, 1, 25}] (* Jean-François Alcover, Feb 07 2017, after Alois P. Heinz *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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