A159571 Fifth diagonal in Table A159572; previous diagonals are A000012, A000217, A000330 and A086602.
6, 39, 138, 364, 804
Offset: 1
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The array begins 1 2 4 7 1 11 5 16 14 2 22 30 12 29 55 39 5 37 91 95 32 1
a(1) = 2 because 4+4+4 and 3+3+3+3 cannot be permuted. a(2) = 41 because there are 3 + 7 + 12 + 9 + 10 ways of permuting the associated partitions. 5553 (3 ways), 4441 & 544 (4+3 ways), 4432 (12 ways), 33331 & 4333 (5 + 4 ways) and 33322 (in 10 ways).
LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{2,41,292,1283,4253,11712,28261,61738,124763},30] (* Harvey P. Dale, Aug 21 2024 *)
a(1) = 1 because the only associated partition 4444 for n = 16 cannot be permuted. a(2) = 64 because the associated partitions can be permuted in 3 + 4 + 12 + 9 + 20 + 10 + 6 ways when n = 17.
The associated sequences begin for n = 15 through 19: ........................1.......5 ........................3.......18 ................3.......18......60 ........3.......18......60......150 1.......7.......25......65......140 ........................6.......42 ................12......84......324 ........12......84......324.....924 ........6.......42......162.....462 4.......32......132.....392.....952 ........................10......80 ................30......240.....1050 ........10......90......420.....1400 ........30......240.....1050....3360 1.......11......56......196.....546 ........................15......135 ................60......540.....2640 ........15......165.....900.....3420 ........................21......210 ................35......385.....2205 ........................28......308 ........................1.......19 therefore this sequence begins 6 126 992 4921 18450
An examination of the relevant ordered Gaussian polynomials reveals the following distribution (beginning with partitions of length three): 1 10 15 1 6 52 180 216 28 12 114 530 1386 1547 266 18 168 880 3086 7007 7616 1554 therefore (by summing each row) this sequence begins 27 482 3855 20329
LinearRecurrence[{13,-78,286,-715,1287,-1716,1716,-1287,715,-286,78,-13,1}, {27,482, 3855,20329,82346,277295,813738,2145712,5192450,11708366,24881487, 50269005, 97217758,180966915,325691821}, 1001] (* Georg Fischer, Feb 28 2019 *)
An examination of the relevant ordered Gaussian polynomials reveals the following distributions: 5 4 7 120 120 34 3 112 1127 1190 470 96 9 882 6692 7147 3270 910 162 15 therefore the sequence begins 9 284 3004 19078 ...
LinearRecurrence[{14,-91,364,-1001,2002,-3003,3432,-3003,2002,-1001,364,-91,14,-1}, {9, 284, 3004,19078, 88938, 335612, 1084387, 3109060, 8104089,19539904, 44141520, 94346102, 192252586, 375787005, 708083995, 1291443529, 2287680232}, 1001] (* Georg Fischer, Feb 28 2019 *)
A110624(1) = 2 because the two relevant partitions of 20 are 4+4+4+4+4 and 5+5+5+5 and cannot be permuted. A110624(2) = 148 because at n = 21 the relevant partitions can be permuted in 20 plus 128 = 148 ways. The 20 ways are apparent from A034261 by observing that 9 + 11 = 20 in the subsequences 1,9,39,... and 1,11,56,... and the 128 ways result from examining the nine partitions of n = 21 having 3,31,32,33,7,211,221,311 and 411 degrees. A110624(4)= A110554(13) = 16746.
The row n=7 starts from the partitions (weights in parentheses) 7 (1), 6+1 (2), 5+2 (2), 4+3 (2), 5+1+1 (3), 4+2+1 (6=3!/1!/1!/1!), 3+3+1 (3), 3+2+2 (3), 4+1+1+1 (4=4!/1!/3!), 3+2+1+1 (12 = 4!/1!/1!/2!), 2+2+2+1 (4), 3+1+1+1+1+1 (5), 2+2+1+1+1 (10=5!/2!/3!), 2+1+1+1+1 (6), 1+1+1+1+1+1 (1). Then T(7,7) = 1+2+3+4+5+6+1 = 22 is the sum of the weights of partitions with largest part 7 and length 1, largest part 6 and length 2,... largest part 1 and length 7. T(7,6) = 2+6+12+10 = 30 is the sum of the weights of the partitions with largest part 6 and length 1, largest part 5 and length 2, ..., largest part 1 and length 6. T(7,5) = 2+3+3+4 = 12 collects all the partitions with largest part 5 and length 1 down to largest part 1 and length 5. The array has A033638(k) nonzero entries per column, starting at n=1 as : 1 ..2 ....4 ....1..7 .......5..11 .......2..14..16 ..........12..30..22 ...........5..39..55..29 ...........1..32..95..91..37 ..............18.113.195.140 ...............7.101.299.357 ...............2
A033638 := proc(n) ( (7+(-1)^n)/2 + n^2 )/4 ; end proc: freq := proc(L,n) local a,p; a := 0 ; for p in L do if p = n then a := a+1 ; end if; end do: a ; end proc: M3 := proc(L) local a,i; a := factorial(nops(L)) ; for i in convert(L,set) do a := a/factorial(freq(L,i)) ; end do: a ; end proc: A105552 := proc(n,k) local p,a,l ; a := 0 ; for p in combinat[partition](n) do if max(op(p)) + nops(p) = k+1 then a := a+ M3(p); end if; end do ; a ; end proc: for k from 1 to 15 do for n from k to k+A033638(k)+1 do T := A105552(n,k) ; if T >0 then printf("%d,", A105552(n,k)) ; end if; end do: printf("\n") ; end do: # R. J. Mathar, Jun 26 2010 # second Maple program: b:= proc(n, k, p) option remember; `if`(n=0 and k=0, 1, `if`(k<1, 0, add(b(n-j, k-1-max(p, j)+p, max(p, j)), j=1..n))) end: T:= k-> seq(b(n, k+1, 0), n=k..k+floor((k-1)^2/4)): seq(T(k), k=1..10); # Alois P. Heinz, Jul 24 2013
b[n_, k_, p_] := b[n, k, p] = If[n == 0 && k == 0, 1, If[k < 1, 0, Sum[b[n - j, k - 1 - Max[p, j] + p, Max[p, j]], {j, 1, n}]]]; T[k_] := Table[b[n, k + 1, 0], {n, k, k + Floor[(k - 1)^2/4]}]; Table[T[k], {k, 1, 10}] // Flatten (* Jean-François Alcover, Jan 08 2016, after Alois P. Heinz *)
The associated partitions begin (for n = 14, 15, 16, ... ................1.......5.......14 ................3.......18......60 ........3.......18......60......150 3.......18......60......150.....315 ................6.......42......162 ........12......84......324.....924 4.......32......132.....392.....952 6.......42......162.....462.....1092 ................10......80......350 ........30......240.....1050....3360 5.......50......245.....840.....2310 ................15......135.....660 ........20......200.....1040....3840 ................21......210.....1134 ................1.......17......125 therefore this sequence begins 18 207 1198 4825 15448 ...
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