A270953
Number T(n,k) of set partitions of [n] having exactly k pairs (m,m+1) such that m+1 is in some block b and m is in block b+1; triangle T(n,k), n>=0, 0<=k<=n-floor((1+sqrt(max(0,8n-7)))/2), read by rows.
Original entry on oeis.org
1, 1, 2, 4, 1, 9, 6, 25, 24, 3, 84, 91, 27, 1, 323, 374, 159, 21, 1377, 1699, 857, 197, 10, 6412, 8410, 4726, 1421, 174, 4, 32312, 44794, 27385, 9573, 1783, 127, 1, 174941, 254718, 167097, 64724, 15158, 1856, 76, 1011357, 1538027, 1071422, 449567, 121464, 20074, 1650, 36
Offset: 0
T(3,1) = 1: 13|2.
T(4,1) = 6: 124|3, 134|2, 13|24, 13|2|4, 14|23, 1|24|3.
T(5,2) = 3: 135|24, 13|25|4, 15|24|3.
T(6,3) = 1: 136|25|4.
T(7,3) = 21: 1247|36|5, 1347|26|5, 1357|246, 135|247|6, 137|246|5, 1367|25|4, 136|257|4, 136|25|47, 136|25|4|7, 137|256|4, 13|257|46, 13|25|47|6, 137|26|45, 13|27|46|5, 147|236|5, 157|246|3, 15|247|36, 15|24|37|6, 17|246|35, 1|247|36|5, 17|26|35|4.
T(8,4) = 10: 1358|247|6, 1368|257|4, 136|258|47, 136|25|48|7, 138|257|46, 13|258|47|6, 138|27|46|5, 158|247|36, 15|248|37|6, 18|247|36|5.
T(9,5) = 4: 1369|258|47, 136|259|48|7, 139|258|47|6, 159|248|37|6.
T(10,6) = 1: 136(10)|259|48|7.
Triangle T(n,k) begins:
00 : 1;
01 : 1;
02 : 2;
03 : 4, 1;
04 : 9, 6;
05 : 25, 24, 3;
06 : 84, 91, 27, 1;
07 : 323, 374, 159, 21;
08 : 1377, 1699, 857, 197, 10;
09 : 6412, 8410, 4726, 1421, 174, 4;
10 : 32312, 44794, 27385, 9573, 1783, 127, 1;
Columns k=0-10 give:
A270954,
A270955,
A270956,
A270957,
A270958,
A270959,
A270960,
A270961,
A270962,
A270963,
A270964.
-
b:= proc(n, i, m) option remember; expand(`if`(n=0, 1, add(
b(n-1, j, max(m, j))*`if`(j=i-1, x, 1), j=1..m+1)))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 1, 0)):
seq(T(n), n=0..14);
-
b[n_, i_, m_] := b[n, i, m] = Expand[If[n == 0, 1, Sum[b[n - 1, j, Max[m, j]]*If[j == i - 1, x, 1], {j, 1, m + 1}]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, 1, 0]]; Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Dec 12 2016, after Alois P. Heinz *)
A271270
Number of set partitions of [n] such that for each pair of consecutive blocks (b,b+1) at least one pair of consecutive numbers (i,i+1) exists with i member of b and i+1 member of b+1.
Original entry on oeis.org
1, 1, 2, 5, 14, 43, 145, 536, 2157, 9371, 43630, 216397, 1137703, 6313675, 36848992, 225464838, 1442216870, 9620746697, 66781675113, 481413175433, 3597627996006, 27825925290597, 222422033403527, 1834910286704787, 15603508329713182, 136616625732498989
Offset: 0
A000110(4) - a(4) = 15 - 14 = 1: 13|2|4.
A000110(5) - a(5) = 52 - 43 = 9: 124|3|5, 134|2|5, 135|2|4, 13|25|4, 13|2|45, 13|2|4|5, 14|23|5, 14|2|3|5, 1|24|3|5.
-
b:= proc(n, i, m, l) option remember; `if`(n=0,
`if`({l[], 1}={1}, 1, 0), add(b(n-1, j, max(m, j),
`if`(j=m+1, [l[], `if`(j=i+1, 1, 0)],
`if`(j=i+1, subsop(j=1, l), l))), j=1..m+1))
end:
a:= n-> b(n, 0$2, []):
seq(a(n), n=0..18);
-
b[n_, i_, m_, l_] := b[n, i, m, l] = If[n == 0, If[Union[l, {1}] == {1}, 1, 0], Sum[b[n-1, j, Max[m, j], If[j == m+1, Join[l, If[j == i+1, {1}, {0}] ], If[j == i+1, ReplacePart[l, j -> 1], l]]], {j, 1, m+1}]]; a[n_] := b[n, 0, 0, {}]; Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Jan 30 2017, translated from Maple *)
A185983
Triangle read by rows: number of set partitions of n elements with k circular connectors.
Original entry on oeis.org
1, 1, 0, 1, 0, 1, 1, 0, 3, 1, 1, 0, 8, 4, 2, 1, 1, 20, 15, 14, 1, 1, 6, 53, 61, 68, 11, 3, 1, 25, 159, 267, 295, 97, 32, 1, 1, 93, 556, 1184, 1339, 694, 242, 28, 3, 1, 346, 2195, 5366, 6620, 4436, 1762, 371, 48, 2, 1, 1356, 9413, 25400, 34991, 27497, 12977, 3650, 634, 53, 3
Offset: 0
For a(4,2) = 8, the set partitions are 1/234, 134/2, 124/3, 123/4, 12/34, 14/23, 1/24/3, and 13/2/4.
For a(5,1) = 1, the set partition is 13/25/4.
For a(6,6) = 3, the set partitions are 135/246, 14/25/36, 1/2/3/4/5/6.
Triangle begins:
1;
1, 0;
1, 0, 1;
1, 0, 3, 1;
1, 0, 8, 4, 2;
1, 1, 20, 15, 14, 1;
1, 6, 53, 61, 68, 11, 3;
...
-
b:= proc(n, i, m, t) option remember; `if`(n=0, x^(t+
`if`(i=m and m<>1, 1, 0)), add(expand(b(n-1, j,
max(m, j), `if`(j=m+1, 0, t+`if`(j=1 and i=m
and j<>m, 1, 0)))*`if`(j=i+1, x, 1)), j=1..m+1))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..n))(b(n, 1, 0$2)):
seq(T(n), n=0..12); # Alois P. Heinz, Mar 30 2016
-
b[n_, i_, m_, t_] := b[n, i, m, t] = If[n == 0, x^(t + If[i == m && m != 1, 1, 0]), Sum[Expand[b[n - 1, j, Max[m, j], If[j == m + 1, 0, t + If[j == 1 && i == m && j != m, 1, 0]]]*If[j == i + 1, x, 1]], {j, 1, m + 1}]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, n}]][b[n, 1, 0, 0] ];
Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, May 19 2016, after Alois P. Heinz *)
A272064
Number of set partitions of [n] such that for each pair of consecutive blocks (b,b+1) exactly one pair of consecutive numbers (i,i+1) exists with i member of b and i+1 member of b+1.
Original entry on oeis.org
1, 1, 2, 5, 13, 35, 102, 332, 1205, 4796, 20640, 95197, 467694, 2435804, 13394117, 77490260, 470198899, 2984034004, 19757370537, 136171758636, 975002124101, 7239322944625, 55648169854405, 442195755123607, 3627392029179270, 30679238282421267, 267215329668444337
Offset: 0
A000110(4) - a(4) = 15 - 13 = 2: 13|24, 13|2|4.
A000110(5) - a(5) = 52 - 35 = 17: 124|35, 124|3|5, 134|25, 134|2|5, 135|24, 13|245, 13|24|5, 135|2|4, 13|25|4, 13|2|45, 13|2|4|5, 14|235, 14|23|5, 14|25|3, 14|2|3|5, 1|24|35, 1|24|3|5.
-
b:= proc(n, i, m, l) option remember; `if`(n=0,
`if`({l[], 1}={1}, 1, 0), add(`if`(j b(n, 0$2, []):
seq(a(n), n=0..18);
-
b[n_, i_, m_, l_] := b[n, i, m, l] = If[n==0, If[Union[Append[l, 1]] == {1}, 1, 0], Sum[If[j 1], l]]]], {j, 1, m+1}]]; a[n_] := b[n, 0, 0, {}]; Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Feb 03 2017, translated from Maple *)
A271271
Number of set partitions of [n] such that at least one pair of consecutive blocks (b,b+1) exists having no pair of consecutive numbers (i,i+1) with i member of b and i+1 member of b+1.
Original entry on oeis.org
0, 0, 0, 0, 1, 9, 58, 341, 1983, 11776, 72345, 462173, 3075894, 21330762, 154050330, 1157493707, 9037925277, 73244123107, 615295131046, 5351329029624, 48126530239366, 447043890866154, 4284293705043796, 42317095568379559, 430355360965092107, 4501973706497500364
Offset: 0
a(4) = 1: 13|2|4.
a(5) = 9: 124|3|5, 134|2|5, 135|2|4, 13|25|4, 13|2|45, 13|2|4|5, 14|23|5, 14|2|3|5, 1|24|3|5.
-
b:= proc(n, i, m, l) option remember; `if`(n=0,
`if`(l=[] or {l[]}={1}, 1, 0), add(b(n-1, j, max(m, j),
`if`(j=m+1, `if`(j=i+1, [l[],1], [l[],0]),
`if`(j=i+1, subsop(j=1, l), l))), j=1..m+1))
end:
a:= n-> combinat[bell](n)-b(n, 0$2, []):
seq(a(n), n=0..18);
-
b[n_, i_, m_, l_] := b[n, i, m, l] = If[n == 0, If[Union[l, {1}] == {1}, 1, 0], Sum[b[n-1, j, Max[m, j], If[j == m+1, Join[l, If[j == i+1, {1}, {0}] ], If[j == i+1, ReplacePart[l, j -> 1], l]]], {j, 1, m+1}]]; a[n_] := BellB[n] - b[n, 0, 0, {}]; Table[a[n], {n, 0, 18}] (* Jean-François Alcover, Feb 02 2017, translated from Maple *)
A271206
Number T(n,k) of set partitions of [n] having exactly k triples (t,t+1,t+2) such that t+i is in block b+i for some b; triangle T(n,k), n>=0, 0<=k<=max(0,n-2), read by rows.
Original entry on oeis.org
1, 1, 2, 4, 1, 10, 4, 1, 28, 18, 5, 1, 89, 77, 30, 6, 1, 315, 345, 164, 45, 7, 1, 1233, 1617, 919, 299, 63, 8, 1, 5285, 8003, 5262, 2011, 492, 84, 9, 1, 24583, 41871, 31180, 13611, 3857, 754, 108, 10, 1, 123062, 231474, 191889, 94020, 30128, 6755, 1095, 135, 11, 1
Offset: 0
T(3,1) = 1: 1|2|3.
T(4,1) = 4: 12|3|4, 14|2|3, 1|24|3, 1|2|34.
T(5,1) = 18: 123|4|5, 125|3|4, 12|35|4, 12|3|45, 13|24|5, 1|23|4|5, 145|2|3, 14|25|3, 14|2|35, 14|2|3|5, 15|24|3, 1|245|3, 1|24|35, 1|24|3|5, 15|2|34, 1|25|34, 1|2|345, 1|2|34|5.
T(5,2) = 5: 12|3|4|5, 15|2|3|4, 1|25|3|4, 1|2|35|4, 1|2|3|45.
T(5,3) = 1: 1|2|3|4|5.
Triangle T(n,k) begins:
: 0 : 1;
: 1 : 1;
: 2 : 2;
: 3 : 4, 1;
: 4 : 10, 4, 1;
: 5 : 28, 18, 5, 1;
: 6 : 89, 77, 30, 6, 1;
: 7 : 315, 345, 164, 45, 7, 1;
: 8 : 1233, 1617, 919, 299, 63, 8, 1;
: 9 : 5285, 8003, 5262, 2011, 492, 84, 9, 1;
: 10 : 24583, 41871, 31180, 13611, 3857, 754, 108, 10, 1;
-
b:= proc(n, i, t, m) option remember; expand(`if`(n=0, 1, add((v->
`if`(t and v, x, 1)*b(n-1, j, v, max(m, j)))(j=i+1), j=1..m+1)))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 1, false, 0)):
seq(T(n), n=0..14);
-
b[n_, i_, t_, m_] := b[n, i, t, m] = Expand[If[n==0, 1, Sum[Function[v, If[t && v, x, 1]*b[n-1, j, v, Max[m, j]]][j==i+1], {j, 1, m+1}]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, 1, False, 0]]; Table[T[n], {n, 0, 14}] // Flatten (* Jean-François Alcover, Feb 05 2017, translated from Maple *)
A271788
Number of set partitions of [n] having exactly one pair (m,m+1) such that m is in some block b and m+1 is in block b+1.
Original entry on oeis.org
0, 1, 3, 7, 16, 39, 105, 314, 1035, 3723, 14494, 60670, 271544, 1293147, 6523495, 34724247, 194357190, 1140402612, 6995760364, 44760085240, 298054873358, 2061644525813, 14787185811993, 109804829195145, 842928183558160, 6680572760715182, 54595535222727960
Offset: 1
a(2) = 1: 1|2.
a(3) = 3: 12|3, 13|2, 1|23.
a(4) = 7: 123|4, 124|3, 12|34, 134|2, 13|2|4, 14|23, 1|234.
a(5) = 16: 1234|5, 1235|4, 123|45, 1245|3, 124|3|5, 125|34, 12|345, 1345|2, 134|2|5, 135|2|4, 13|25|4, 13|2|45, 145|23, 14|23|5, 15|234, 1|2345.
-
b:= proc(n, i, m, k) option remember; `if`(n=0, `if`(k=0, 1, 0),
add(`if`(j=i+1 and k=0, 0, b(n-1, j, max(m, j), k-
`if`(j=i+1, 1, 0))), j=1..m+1))
end:
a:= n-> b(n, 1, 0, 1):
seq(a(n), n=1..30);
-
b[n_, i_, m_, k_] := b[n, i, m, k] = If[n == 0, If[k == 0, 1, 0], Sum[If[j == i + 1 && k == 0, 0, b[n - 1, j, Max[m, j], k - If[j == i + 1, 1, 0]]], {j, 1, m + 1}]];
a[n_] := b[n, 1, 0, 1];
Table[a[n], {n, 1, 30}] (* Jean-François Alcover, May 27 2018, translated from Maple *)
A271789
Number of set partitions of [n] having exactly two pairs (m,m+1) such that m is in some block b and m+1 is in block b+1.
Original entry on oeis.org
0, 1, 6, 24, 86, 307, 1143, 4513, 18956, 84546, 399218, 1989792, 10439521, 57504306, 331747730, 1999924893, 12571830681, 82245682149, 558951842996, 3939679356783, 28754596249395, 217019729585609, 1691485555633721, 13598390501982510, 112633410446366669
Offset: 2
a(3) = 1: 1|2|3.
a(4) = 6: 12|3|4, 13|24, 1|23|4, 14|2|3, 1|24|3, 1|2|34.
a(5) = 24: 123|4|5, 124|35, 12|34|5, 125|3|4, 12|35|4, 12|3|45, 134|25, 135|24, 13|245, 13|2|4|5, 14|235, 1|234|5, 15|23|4, 1|235|4, 1|23|45, 145|2|3, 14|2|35, 14|2|3|5, 15|24|3, 1|245|3, 1|24|3|5, 15|2|34, 1|25|34, 1|2|345.
A271790
Number of set partitions of [n] having exactly three pairs (m,m+1) such that m is in some block b and m+1 is in block b+1.
Original entry on oeis.org
0, 1, 10, 61, 313, 1520, 7373, 36627, 188396, 1007708, 5612677, 32551400, 196447095, 1232448403, 8028591686, 54240974862, 379574487845, 2748022434268, 20558256639096, 158746877185099, 1263892981174760, 10364745848886935, 87464284123646375, 758804198576946316
Offset: 3
a(4) = 1: 1|2|3|4.
a(5) = 10: 12|3|4|5, 13|24|5, 1|23|4|5, 14|25|3, 1|24|35, 1|2|34|5, 15|2|3|4, 1|25|3|4, 1|2|35|4, 1|2|3|45.
A271791
Number of set partitions of [n] having exactly four pairs (m,m+1) such that m is in some block b and m+1 is in block b+1.
Original entry on oeis.org
0, 1, 15, 129, 891, 5611, 34213, 208230, 1285094, 8108722, 52540098, 350375379, 2407395908, 17048774736, 124435165429, 935736058614, 7246161094614, 57749473020305, 473358586652328, 3987856784525223, 34506189729577186, 306454001139880689, 2791593349164080381
Offset: 4
a(5) = 1: 1|2|3|4|5.
a(6) = 15: 12|3|4|5|6, 13|24|5|6, 1|23|4|5|6, 14|25|36, 1|24|35|6, 1|2|34|5|6, 15|26|3|4, 1|25|36|4, 1|2|35|46, 1|2|3|45|6, 16|2|3|4|5, 1|26|3|4|5, 1|2|36|4|5, 1|2|3|46|5, 1|2|3|4|56.
Showing 1-10 of 20 results.
Comments