A209936
Triangle of multiplicities of k-th partition of n corresponding to sequence A080577. Multiplicity of a given partition of n into k parts is the number of ways parts can be selected from k distinguishable bins. See the example.
Original entry on oeis.org
1, 2, 1, 3, 6, 1, 4, 12, 6, 12, 1, 5, 20, 20, 30, 30, 20, 1, 6, 30, 30, 60, 15, 120, 60, 20, 90, 30, 1, 7, 42, 42, 105, 42, 210, 140, 105, 105, 420, 105, 140, 210, 42, 1, 8, 56, 56, 168, 56, 336, 280, 28, 336, 168, 840, 280, 168, 420, 840, 1120, 168, 70, 560, 420, 56, 1
Offset: 1
Triangle begins:
1
2, 1
3, 6, 1
4, 12, 6, 12, 1
5, 20, 20, 30, 30, 20, 1
6, 30, 30, 60, 15, 120, 60, 20, 90, 30, 1
7, 42, 42, 105, 42, 210, 140, 105, 105, 420, 105, 140, 210, 42, 1
...
Thus for n=3 (third row) the partitions of n=3 are:
3+0+0 0+3+0 0+0+3 (multiplicity=3),
2+1+0 2+0+1 1+2+0 1+0+2 0+2+1 0+1+2 (multiplicity=6),
1+1+1 (multiplicity=1).
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Apply[Multinomial,Last/@Tally[#]&/@PadRight[IntegerPartitions[n]],1] (* Wouter Meeussen, Jan 26 2025 *)
A210238
Triangle of multiplicities D(n) of multinomial coefficients corresponding to sequence A210237.
Original entry on oeis.org
1, 2, 1, 3, 6, 1, 4, 12, 6, 12, 1, 5, 20, 20, 30, 30, 20, 1, 6, 30, 30, 15, 60, 120, 20, 60, 90, 30, 1, 7, 42, 42, 42, 105, 210, 105, 245, 420, 140, 105, 210, 42, 1, 8, 56, 56, 224, 28, 336, 336, 280, 168, 168, 840, 420, 1120, 70, 1120, 560, 168, 420, 56, 1
Offset: 1
1
2, 1
3, 6, 1
4, 12, 6, 12, 1
5, 20, 20, 30, 30, 20, 1
6, 30, 30, 15, 60, 120, 20, 60, 90, 30, 1
7, 42, 42, 42, 105, 210, 105, 245, 420, 140, 105, 210, 42, 1
Thus for n=3 (third row) the same value of multinomial coefficient follows from the following combinations:
3!/(3!0!0!) 3!/(0!3!0!) 3!/(0!0!3!) (i.e. multiplicity=3)
3!/(2!1!0!) 3!/(2!0!1!) 3!/(0!2!1!) 3!/(0!1!2!) 3!/(1!0!2!) 3!/(1!2!0!) (i.e. multiplicity=6)
3!/(1!1!1!) (i.e. multiplicity=1)
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Table[Last/@Tally[Multinomial@@@Compositions[k,k]],{k,8}] (* Wouter Meeussen, Mar 09 2013 *)
A214308
a(n) is the number of all two colored bracelets (necklaces with turning over allowed) with n beads with the two colors from a repertoire of n distinct colors, for n>=2.
Original entry on oeis.org
1, 6, 24, 60, 165, 336, 784, 1584, 3420, 6820, 14652, 29484, 62335, 128310, 269760, 558960, 1175499, 2446668, 5131900, 10702020, 22385517, 46655224, 97344096, 202555800, 421478200, 875297124, 1816696728, 3764747868, 7795573230, 16121364000, 33310887808
Offset: 2
a(5) = A213941(5,2) + A213941(5,3) = 20 + 40 = 60 from the bracelet (with colors j for c[j], j=1,2,..,5) cyclic(11112) which represents a class of order A035206(5,2) = 20 (if all 5 colors are used), cyclic(11122) and cyclic(11212) each representing also a color class of 20 members each, summing to 60 bracelets with five beads and five colors available for the two color signatures [4,1] and [3,2].
A130273
Refines A075197(n): number of partitions of n balls of n colors. The refinement has shape A000041(n).
Original entry on oeis.org
1, 4, 2, 9, 24, 5, 20, 84, 54, 132, 15, 35, 240, 320, 630, 780, 720, 52, 66, 570, 870, 2280, 465, 6240, 4440, 1320, 8280, 4050, 203, 105, 1260, 1974, 6720, 2394, 20580, 19740, 14385, 11445, 83160, 31080, 34860, 77910, 23772, 877, 176, 2520, 4312, 17640, 5432
Offset: 1
The array begins
1
4 2
9 24 5
20 84 54 132 15
...
Row three is (9,24,5) because there are (3, 4,5) cases; and we have (3, 6,1) ways to pick 1,2 or 3 colors.
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