A249620
Triangle read by rows: T(m,n) = number of partitions of the multiset with m elements and signature corresponding to n-th integer partition (A194602).
Original entry on oeis.org
1, 1, 2, 2, 5, 4, 3, 15, 11, 7, 9, 5, 52, 36, 21, 26, 12, 16, 7, 203, 135, 74, 92, 38, 52, 19, 66, 29, 31, 11, 877, 566, 296, 371, 141, 198, 64, 249, 98, 109, 30, 137, 47, 57, 15, 4140, 2610, 1315, 1663, 592, 850, 250, 1075, 392, 444, 105, 560
Offset: 0
See "The T(5,2)=21 partitions of {1,1,1,2,3}" link. Similar links for m=1..8 are in "Partitions of multisets" (Wikiversity).
Triangle begins:
n 0 1 2 3 4 5 6 7 8 9 10
m
0 1
1 1
2 2 2
3 5 4 3
4 15 11 7 9 5
5 52 36 21 26 12 16 7
6 203 135 74 92 38 52 19 66 29 31 11
- Tilman Piesk, Triangle rows m=0..8, flattened.
- Tilman Piesk, Partitions of multisets (Wikiversity)
- Tilman Piesk, The T(5,2)=21 partitions of {1,1,1,2,3}
- Tilman Piesk, PHP code used to calculate the examples
A093936
Table T(n,k) read by rows which contains in row n and column k the sum of A001055(A036035(n,j)) over all column indices j where A036035(n,j) has k distinct prime factors.
Original entry on oeis.org
1, 2, 2, 3, 4, 5, 5, 16, 11, 15, 7, 28, 47, 36, 52, 11, 79, 156, 166, 135, 203, 15, 134, 408, 588, 667, 566, 877, 22, 328, 1057, 2358, 2517, 2978, 2610, 4140, 30, 536, 3036, 6181, 10726, 11913, 14548, 13082, 21147, 42, 1197, 6826, 21336, 40130, 53690, 61421
Offset: 1
a(19) = 166 because A001055(840) + A001055(1260) = 74 + 92.
Row n=4 of A036035 contains 16=2^4, 24=2^3*3, 36=2^2*3^2, 60=2^2*3*5 and 210=2*3*5*7. The 16 has k=1 distinct prime factor; 24 and 36 have k=2 distinct prime factors; 60 has k=3 distinct prime factors; 210 has k=4 distinct prime factors (see A001221).
T(4,1)=A001055(16)=5.
T(4,2)=A001055(24)+A001055(36)=7+9=16.
T(4,3)=A001055(60)=11.
T(4,4)=A001055(210)=15.
Table starts
1;
2, 2;
3, 4, 5;
5, 16, 11, 15;
7, 28, 47, 36, 52;
11, 79, 156, 166, 135, 203;
15, 134, 408, 588, 667, 566, 877;
22, 328, 1057, 2358, 2517, 2978, 2610, 4140;
30, 536, 3036, 6181, 10726, 11913, 14548, 13082, 21147;
42, 1197, 6826, 21336, 40130, 53690, 61421, 76908, 70631, 115975;
...
-
A036035 := proc(n) local pr,L,a ; a := [] ; pr := combinat[partition](n) ; for L in pr do mul(ithprime(i)^op(-i,L),i=1..nops(L)) ; a := [op(a),%] ; od ; RETURN(a) ; end: A001221 := proc(n) local ifacts ; ifacts := ifactors(n)[2] ; nops(ifacts) ; end: listProdRep := proc(n,mincomp) local dvs,resul,f,i,rli ; resul := 0 ; if n = 1 then RETURN(1) elif n >= mincomp then dvs := numtheory[divisors](n) ; for i from 1 to nops(dvs) do f := op(i,dvs) ; if f =n and f >= mincomp then resul := resul+1 ; elif f >= mincomp then rli := listProdRep(n/f,f) ; resul := resul+rli ; fi ; od ; fi ; RETURN(resul) ; end: A001055 := proc(n) listProdRep(n,2) ; end: A093936 := proc(n,k) local a, a036035,j ; a := 0 ; a036035 := A036035(n) ; for j in a036035 do if A001221(j) = k then a := a+A001055(j) ; fi ; od ; RETURN(a) ; end: for n from 1 to 10 do for k from 1 to n do printf("%d,",A093936(n,k)) ; od : od : # R. J. Mathar, Jul 27 2007
A129305
Encodes multisets of least prime signatures in reverse-lex order: replace A036035 with A080688 then calculate all possible factorizations of the resulting values, recode each factor using A064553(n) and then multiply the terms.
Original entry on oeis.org
1, 2, 4, 5, 6, 11, 8, 10, 17, 12, 15, 22, 31, 42, 69, 77, 86, 109, 16, 20, 25, 34, 47, 24, 30, 44, 55, 51, 62, 83, 36, 45, 66, 76, 95, 121, 93, 118, 149, 84, 105, 138, 154, 172, 215, 253, 201, 217, 218, 277, 546, 834, 861, 897, 994, 1001, 1118, 1529, 1633, 1763, 1041
Offset: 0
The encoded values can be arranged in tabular form based on the number of factors and the associated numeric partitions as indicated below:
2..................................................
.....4.....5........................................
.....6.....11........................................
...............8.....10.....17.........................
...............12....15.....31.........................
.....................22..............................
...............42....69.....109.........................
.....................77..............................
.....................86..............................
.................................16.....20.....25.....34.....47
.................................24.....30.....55.....51.....83
........................................44............62.....
.................................36.....45.....95.....93.....149
........................................66.....121...118.....
........................................76...............
.................................84.....105.....215.....201.....277
........................................138.....253.....217.....
........................................154.............218.....
........................................172...............
................................546.....834.....1529.....1041.....1289
........................................861.....1633.....1138.....
........................................897.....1763.....1253.....
........................................994..............1417.....
........................................1001...............
........................................1118...............
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.
Comments