A240796
Total number of occurrences of the pattern 1<2 in all preferential arrangements (or ordered partitions) of n elements.
Original entry on oeis.org
0, 1, 15, 186, 2330, 31065, 447405, 6979588, 117745668, 2141106795, 41810587775, 873474855726, 19451904450654, 460209050303821, 11531197020389025, 305122289460210120, 8503747639606509128, 249020038061419770783, 7645072502094118876755, 245564189847880300238290
Offset: 1
The 13 preferential arrangements on 3 points and the number of times the pattern 1<2 occurs are:
1<2<3, 3
1<3<2, 2
2<1<3, 2
2<3<1, 1
3<1<2, 1
3<2<1, 0
1=2<3, 2
1=3<2, 1
2=3<1, 0
1<2=3, 2
2<1=3, 1
3<1=2, 0
1=2=3, 0,
for a total of a(3) = 15.
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b:= proc(n, t) option remember; `if`(n=0, [1, 0], add((p-> p+
[0, p[1]*j*t/2])(b(n-j, t+j))*binomial(n, j), j=1..n))
end:
a:= n-> b(n, 0)[2]:
seq(a(n), n=1..25); # Alois P. Heinz, Dec 08 2014
-
b[n_, t_] := b[n, t] = If[n == 0, {1, 0}, Sum[Function[{p}, p + {0, p[[1]]*j*t/2}][b[n - j, t + j]]*Binomial[n, j], {j, 1, n}]]; a[n_] := b[n, 0][[2]]; Table[a[n], {n, 1, 25}] (* Jean-François Alcover, Jun 08 2015, after Alois P. Heinz *)
A240797
Total number of occurrences of the pattern 1=2 in all preferential arrangements (or ordered partitions) of n elements.
Original entry on oeis.org
0, 1, 9, 78, 750, 8115, 98343, 1324204, 19650060, 318926745, 5623615965, 107093749818, 2191142272410, 47944109702671, 1117341011896515, 27633982917342360, 722929036749464280, 19946727355457792853, 578926427416920550233, 17632301590672398115270
Offset: 1
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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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]:
seq(a(n), n=1..25); # Alois P. Heinz, Dec 08 2014
-
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 *)
A240798
Total number of occurrences of the pattern 1=2=3 in all preferential arrangements (or ordered partitions) of n elements.
Original entry on oeis.org
0, 0, 1, 12, 130, 1500, 18935, 262248, 3972612, 65500200, 1169398065, 22494463860, 464072915878, 10225330604580, 239720548513355, 5959152063448080, 156592569864940040, 4337574220496785680, 126329273251232688069, 3859509516112803668220, 123426111134706786806890
Offset: 1
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b:= proc(n) option remember; `if`(n=0, [1, 0], add((p-> p+
[0, p[1]*binomial(j, 3)])(b(n-j))*binomial(n, j), j=1..n))
end:
a:= n-> b(n)[2]:
seq(a(n), n=1..25); # Alois P. Heinz, Dec 08 2014
-
b[n_] := b[n] = If[n==0, {1, 0}, Sum[Function[p, p+{0, p[[1]]*Binomial[j, 3]} ][b[n-j]]*Binomial[n, j], {j, 1, n}]]; a[n_] := b[n][[2]]; Table[a[n], {n, 1, 25}] (* Jean-François Alcover, Feb 28 2017, after Alois P. Heinz *)
A240799
Total number of occurrences of the pattern 1=2<3 in all preferential arrangements (or ordered partitions) of n elements.
Original entry on oeis.org
0, 0, 1, 20, 310, 4660, 72485, 1193080, 20938764, 392485560, 7850724915, 167242351100, 3785057708146, 90775554103052, 2301045251519105, 61499717442074800, 1729026306941190680, 51022485837639054768, 1577126907722325214959, 50967150013960792511700
Offset: 1
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b:= proc(n, t) option remember; `if`(n=0, [1, 0], add((p-> p+
[0, p[1]*j*(j-1)*t/6])(b(n-j, t+j))*binomial(n, j), j=1..n))
end:
a:= n-> b(n, 0)[2]:
seq(a(n), n=1..25); # Alois P. Heinz, Dec 08 2014
-
b[n_, t_] := b[n, t] = If[n==0, {1, 0}, Sum[Function[p, p+{0, p[[1]]*j*(j-1)*t/6}][b[n-j, t+j]]*Binomial[n, j], {j, 1, n}]]; a[n_] := b[n, 0][[2]]; Table[a[n], {n, 1, 25}] (* Jean-François Alcover, Feb 28 2017, after Alois P. Heinz *)
Showing 1-4 of 4 results.
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