A200660
Sum of the number of arcs describing the set partitions of {1,2,...,n}.
Original entry on oeis.org
0, 1, 8, 49, 284, 1658, 9974, 62375, 406832, 2769493, 19668054, 145559632, 1121153604, 8974604065, 74553168520, 641808575961, 5718014325296, 52653303354906, 500515404889978, 4905937052293759, 49530189989912312, 514541524981377909, 5494885265473192914
Offset: 1
- Seiichi Manyama, Table of n, a(n) for n = 1..500
- M. Aguiar, C. Andre, C. Benedetti, N. Bergeron, Z. Chen, P. Diaconis, A. Hendrickson, S. Hsiao, I. M. Isaacs, A. Jedwab, K. Johnson, G. Karaali, A. Lauve, T. Le, S. Lewis, H. Li, K. Magaard, E. Marberg, J-C. Novelli, A. Pang, F. Saliola, L. Tevlin, J-Y. Thibon, N. Thiem, V. Venkateswaran, C. R. Vinroot, N. Yan, and M. Zabrocki, Supercharacters, symmetric functions in noncommuting variables, and related Hopf algebras, arXiv:1009.4134 [math.CO], 2010-2011.
- C. André, Basic characters of the unitriangular group, Journal of Algebra, 175 (1995), 287-319.
Cf.
A011971 (sequence is computed from Aitken's array b(n,k) arcs(n) = Sum_{k=1..n-1} k*b(n,k)).
Cf.
A200580,
A200673 (other statistics related to supercharacter table).
-
b:=proc(n,k) option remember;
if n=1 and k=1 then RETURN(1) fi;
if k=1 then RETURN(b(n-1,n-1)) fi;
b(n,k-1)+b(n-1,k-1)
end:
arcs:=proc(n) local res,k;
res:=0;
for k to n-1 do res:=res+ k*b(n,k) od;
res
end:
seq(arcs(n),n=1..34);
-
b[n_, k_] := b[n, k] = Which[n == 1 && k == 1, 1, k == 1, b[n - 1, n - 1], True, b[n, k - 1] + b[n - 1, k - 1]];
arcs[n_] := Module[{res = 0, k}, For[k = 1, k <= n-1, k++, res = res + k * b[n, k]]; res];
Array[arcs, 34] (* Jean-François Alcover, Nov 25 2017, translated from Maple *)
A200673
Total number of nested arcs in the set partitions of n.
Original entry on oeis.org
0, 0, 0, 1, 16, 170, 1549, 13253, 110970, 928822, 7862353, 67758488, 596837558, 5385257886, 49837119320, 473321736911, 4614233950422, 46168813528478, 474017189673555, 4992024759165631, 53902161267878974, 596448192670732180, 6760141422115666131, 78438566784031690720
Offset: 1
- M. Aguiar, C. Andre, C. Benedetti, N. Bergeron, Z. Chen, P. Diaconis, A. Hendrickson, S. Hsiao, I.M. Isaacs, A. Jedwab, K. Johnson, G. Karaali, A. Lauve, T. Le, S. Lewis, H. Li, K. Magaard, E. Marberg, J-C. Novelli, A. Pang, F. Saliola, L. Tevlin, J-Y. Thibon, N. Thiem, V. Venkateswaran, C.R. Vinroot, N. Yan, M. Zabrocki, Supercharacters, symmetric functions in noncommuting variables, and related Hopf algebras, arXiv:1009.4134 [math.CO], 2010-2011.
- C. André, Basic characters of the unitriangular group, Journal of Algebra, 175 (1995), 287-319.
Cf.
A200580,
A200660 (other statistics related to supercharacter table).
-
c:=proc(n,k,j) option remember;
if n=3 and k=2 and j=1 then RETURN(1) fi;
if k=2 and j=1 then RETURN(c(n-1,n-2,1)) fi;
if k=j+1 then RETURN(c(n,j+1,j-1) + c(n-1,j,j-1)) fi;
c(n,k-1,j)+c(n-1,k-1,j)
end:
nst:=proc(n) local res,k,j;
res:=0;
for j to n-3 do
for k from j+1 to n-2 do
res:=res+j*(k-j)*c(n,k,j) od; od;
res
end:
seq(nst(n),n=1..21);
-
c[n_, k_, j_] := c[n, k, j] = Which[n == 3 && k == 2 && j == 1, 1, k == 2 && j == 1, c[n - 1, n - 2, 1], k == j + 1, c[n, j + 1, j - 1] + c[n - 1, j, j - 1], True, c[n, k - 1, j] + c[n - 1, k - 1, j]];
nst[n_] := Module[{res = 0, k, j}, For[j = 1, j <= n - 3, j++, For[k = j + 1, k <= n - 2, k++, res = res + j*(k - j)*c[n, k, j]]]; res];
Array[nst, 21] (* Jean-François Alcover, Nov 25 2017, translated from Maple *)
A226507
4*B(n+4) - (4*n+15)*B(n+3) + (n^2+8*n+9)*B(n+2) - (4*n+3)*B(n+1) + n*B(n), where B(i) are the Bell numbers A000110.
Original entry on oeis.org
0, 0, 0, 1, 16, 177, 1726, 15912, 143148, 1279939, 11504326, 104686659, 968808308, 9144180028, 88184565504, 869867691833, 8781919559956, 90765497635245, 960434143555986, 10403548856756708, 115336464546432180, 1308260884070774299, 15177980646442995698, 180036437138753006607, 2182526416321158803528
Offset: 0
-
[4*Bell(n+4)-(4*n+15)*Bell(n+3)+(n^2+8*n+9)*Bell(n+2)-(4*n+3)*Bell(n+1)+n*Bell(n): n in [0..30]]; // Vincenzo Librandi, Jul 16 2013
-
A000110 := proc(n) option remember; if n <= 1 then 1 else add( binomial(n-1, i)*A000110(n-1-i), i=0..n-1); fi; end;
B:=A000110;
f:=n->4*B(n+4) - (4*n+15)*B(n+3) + (n^2+8*n+9)*B(n+2) - (4*n+3)*B(n+1) + n*B(n);
[seq(f(n),n=0..30)];
-
Table[4 BellB[n+4] - (4 n + 15) BellB[n + 3] + (n^2 + 8 n + 9) BellB[n+2] - (4 n + 3) BellB[n+1] + n BellB[n],{n, 0, 30}] (* Vincenzo Librandi, Jul 16 2013 *)
Showing 1-3 of 3 results.
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