The On-Line Encyclopedia of Integer Sequences (OEIS)

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A261275

Number of set partitions C_t(n) of {1,2,...,t} into at most n parts, with an even number of elements in each part distinguished by marks; triangle C_t(n), t>=0, 0<=n<=t, read by rows.
(history; published version)

#25 by Alois P. Heinz at Sat Jun 18 12:28:08 EDT 2022

STATUS

editing

approved

#24 by Alois P. Heinz at Sat Jun 18 12:27:38 EDT 2022

EXAMPLE

1;

0, 1;

0, 2, 3;

0, 4, 10, 11;

0, 8, 36, 48, 49;

0, 16, 136, 236, 256, 257;

0, 32, 528, 1248, 1508, 1538, 1539;

0, 64, 2080, 6896, 9696, 10256, 10298, 10299;

..

...

#23 by Alois P. Heinz at Sat Jun 18 12:26:39 EDT 2022

DATA

1, 0, 1, 0, 2, 3, 0, 4, 10, 11, 0, 8, 36, 48, 49, 0, 16, 136, 236, 256, 257, 0, 32, 528, 1248, 1508, 1538, 1539, 0, 64, 2080, 6896, 9696, 10256, 10298, 10299, 0, 128, 8256, 39168, 66384, 74784, 75848, 75904, 75905, 0, 256, 32896, 226496, 475136, 586352, 607520, 609368, 609440, 609441

STATUS

approved

editing

#22 by Alois P. Heinz at Tue Nov 07 07:23:37 EST 2017

STATUS

proposed

approved

#21 by Jean-François Alcover at Tue Nov 07 06:49:55 EST 2017

STATUS

editing

proposed

#20 by Jean-François Alcover at Tue Nov 07 06:49:45 EST 2017

MATHEMATICA

CC[t_, n_] := Sum[2^(t - m)*StirlingS2[t, m], {m, 0, n}];

Table[CC[t, n], {t, 0, 12}, {n, 0, t}] // Flatten

(* Second program: *)

multinomial[n_, k_List] := n!/Times @@ (k!);

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[x^j*multinomial[n, Join[{n - i*j}, Table[i, j]]]/j!*Sum[Binomial[i, 2*k], {k, 0, i/2}]^j*b[n - i*j, i - 1], {j, 0, n/i}]]];

T[n_] := Function[p, Table[Sum[Coefficient[p, x, j], {j, 0, i}], {i, 0, n} ] ][b[n, n]];

Table[T[n], {n, 0, 12}] // Flatten (* Jean-François Alcover, Nov 07 2017, after Alois P. Heinz *)

STATUS

approved

editing

#19 by Joerg Arndt at Sat Aug 15 02:40:52 EDT 2015

STATUS

proposed

approved

#18 by Mark Wildon at Fri Aug 14 17:03:17 EDT 2015

STATUS

editing

proposed

#17 by Mark Wildon at Fri Aug 14 17:02:20 EDT 2015

COMMENTS

C_t(n) = <pi^t, 1_{BSym_n}> where pi is the permutation character of the hyperoctahedral group BSym_n = C_2 \wr wreath Sym_n given by its imprimitive action on a set of size 2n. This gives a combinatorial interpretation of C_t(n) using sequences of box moves on pairs of Young diagrams.

STATUS

proposed

editing

Discussion

Fri Aug 14

17:03

Mark Wildon: \wr is LaTeX for the wreath product symbol. I've changed it to 'wreath' which I think will be clearer.

#16 by Mark Wildon at Fri Aug 14 16:53:48 EDT 2015

STATUS

editing

proposed

Discussion

Fri Aug 14

17:01

Alois P. Heinz: what is "\wr"?