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Identifier
Values
=>
Cc0002;cc-rep
[]=>1 [1]=>1 [2]=>1 [1,1]=>1 [3]=>1 [2,1]=>3 [1,1,1]=>1 [4]=>1 [3,1]=>4 [2,2]=>2 [2,1,1]=>6 [1,1,1,1]=>1 [5]=>1 [4,1]=>5 [3,2]=>5 [3,1,1]=>10 [2,2,1]=>10 [2,1,1,1]=>10 [1,1,1,1,1]=>1 [6]=>1 [5,1]=>6 [4,2]=>6 [4,1,1]=>15 [3,3]=>3 [3,2,1]=>30 [3,1,1,1]=>20 [2,2,2]=>5 [2,2,1,1]=>30 [2,1,1,1,1]=>15 [1,1,1,1,1,1]=>1 [7]=>1 [6,1]=>7 [5,2]=>7 [5,1,1]=>21 [4,3]=>7 [4,2,1]=>42 [4,1,1,1]=>35 [3,3,1]=>21 [3,2,2]=>21 [3,2,1,1]=>105 [3,1,1,1,1]=>35 [2,2,2,1]=>35 [2,2,1,1,1]=>70 [2,1,1,1,1,1]=>21 [1,1,1,1,1,1,1]=>1 [8]=>1 [7,1]=>8 [6,2]=>8 [6,1,1]=>28 [5,3]=>8 [5,2,1]=>56 [5,1,1,1]=>56 [4,4]=>4 [4,3,1]=>56 [4,2,2]=>28 [4,2,1,1]=>168 [4,1,1,1,1]=>70 [3,3,2]=>28 [3,3,1,1]=>84 [3,2,2,1]=>168 [3,2,1,1,1]=>280 [3,1,1,1,1,1]=>56 [2,2,2,2]=>14 [2,2,2,1,1]=>140 [2,2,1,1,1,1]=>140 [2,1,1,1,1,1,1]=>28 [1,1,1,1,1,1,1,1]=>1 [9]=>1 [8,1]=>9 [7,2]=>9 [7,1,1]=>36 [6,3]=>9 [6,2,1]=>72 [6,1,1,1]=>84 [5,4]=>9 [5,3,1]=>72 [5,2,2]=>36 [5,2,1,1]=>252 [5,1,1,1,1]=>126 [4,4,1]=>36 [4,3,2]=>72 [4,3,1,1]=>252 [4,2,2,1]=>252 [4,2,1,1,1]=>504 [4,1,1,1,1,1]=>126 [3,3,3]=>12 [3,3,2,1]=>252 [3,3,1,1,1]=>252 [3,2,2,2]=>84 [3,2,2,1,1]=>756 [3,2,1,1,1,1]=>630 [3,1,1,1,1,1,1]=>84 [2,2,2,2,1]=>126 [2,2,2,1,1,1]=>420 [2,2,1,1,1,1,1]=>252 [2,1,1,1,1,1,1,1]=>36 [1,1,1,1,1,1,1,1,1]=>1 [10]=>1 [9,1]=>10 [8,2]=>10 [8,1,1]=>45 [7,3]=>10 [7,2,1]=>90 [7,1,1,1]=>120 [6,4]=>10 [6,3,1]=>90 [6,2,2]=>45 [6,2,1,1]=>360 [6,1,1,1,1]=>210 [5,5]=>5 [5,4,1]=>90 [5,3,2]=>90 [5,3,1,1]=>360 [5,2,2,1]=>360 [5,2,1,1,1]=>840 [5,1,1,1,1,1]=>252 [4,4,2]=>45 [4,4,1,1]=>180 [4,3,3]=>45 [4,3,2,1]=>720 [4,3,1,1,1]=>840 [4,2,2,2]=>120 [4,2,2,1,1]=>1260 [4,2,1,1,1,1]=>1260 [4,1,1,1,1,1,1]=>210 [3,3,3,1]=>120 [3,3,2,2]=>180 [3,3,2,1,1]=>1260 [3,3,1,1,1,1]=>630 [3,2,2,2,1]=>840 [3,2,2,1,1,1]=>2520 [3,2,1,1,1,1,1]=>1260 [3,1,1,1,1,1,1,1]=>120 [2,2,2,2,2]=>42 [2,2,2,2,1,1]=>630 [2,2,2,1,1,1,1]=>1050 [2,2,1,1,1,1,1,1]=>420 [2,1,1,1,1,1,1,1,1]=>45 [1,1,1,1,1,1,1,1,1,1]=>1 [11]=>1 [10,1]=>11 [9,2]=>11 [9,1,1]=>55 [8,3]=>11 [8,2,1]=>110 [8,1,1,1]=>165 [7,4]=>11 [7,3,1]=>110 [7,2,2]=>55 [7,2,1,1]=>495 [7,1,1,1,1]=>330 [6,5]=>11 [6,4,1]=>110 [6,3,2]=>110 [6,3,1,1]=>495 [6,2,2,1]=>495 [6,2,1,1,1]=>1320 [6,1,1,1,1,1]=>462 [5,5,1]=>55 [5,4,2]=>110 [5,4,1,1]=>495 [5,3,3]=>55 [5,3,2,1]=>990 [5,3,1,1,1]=>1320 [5,2,2,2]=>165 [5,2,2,1,1]=>1980 [5,2,1,1,1,1]=>2310 [5,1,1,1,1,1,1]=>462 [4,4,3]=>55 [4,4,2,1]=>495 [4,4,1,1,1]=>660 [4,3,3,1]=>495 [4,3,2,2]=>495 [4,3,2,1,1]=>3960 [4,3,1,1,1,1]=>2310 [4,2,2,2,1]=>1320 [4,2,2,1,1,1]=>4620 [4,2,1,1,1,1,1]=>2772 [4,1,1,1,1,1,1,1]=>330 [3,3,3,2]=>165 [3,3,3,1,1]=>660 [3,3,2,2,1]=>1980 [3,3,2,1,1,1]=>4620 [3,3,1,1,1,1,1]=>1386 [3,2,2,2,2]=>330 [3,2,2,2,1,1]=>4620 [3,2,2,1,1,1,1]=>6930 [3,2,1,1,1,1,1,1]=>2310 [3,1,1,1,1,1,1,1,1]=>165 [2,2,2,2,2,1]=>462 [2,2,2,2,1,1,1]=>2310 [2,2,2,1,1,1,1,1]=>2310 [2,2,1,1,1,1,1,1,1]=>660 [2,1,1,1,1,1,1,1,1,1]=>55 [1,1,1,1,1,1,1,1,1,1,1]=>1 [12]=>1 [11,1]=>12 [10,2]=>12 [10,1,1]=>66 [9,3]=>12 [9,2,1]=>132 [9,1,1,1]=>220 [8,4]=>12 [8,3,1]=>132 [8,2,2]=>66 [8,2,1,1]=>660 [8,1,1,1,1]=>495 [7,5]=>12 [7,4,1]=>132 [7,3,2]=>132 [7,3,1,1]=>660 [7,2,2,1]=>660 [7,2,1,1,1]=>1980 [7,1,1,1,1,1]=>792 [6,6]=>6 [6,5,1]=>132 [6,4,2]=>132 [6,4,1,1]=>660 [6,3,3]=>66 [6,3,2,1]=>1320 [6,3,1,1,1]=>1980 [6,2,2,2]=>220 [6,2,2,1,1]=>2970 [6,2,1,1,1,1]=>3960 [6,1,1,1,1,1,1]=>924 [5,5,2]=>66 [5,5,1,1]=>330 [5,4,3]=>132 [5,4,2,1]=>1320 [5,4,1,1,1]=>1980 [5,3,3,1]=>660 [5,3,2,2]=>660 [5,3,2,1,1]=>5940 [5,3,1,1,1,1]=>3960 [5,2,2,2,1]=>1980 [5,2,2,1,1,1]=>7920 [5,2,1,1,1,1,1]=>5544 [5,1,1,1,1,1,1,1]=>792 [4,4,4]=>22 [4,4,3,1]=>660 [4,4,2,2]=>330 [4,4,2,1,1]=>2970 [4,4,1,1,1,1]=>1980 [4,3,3,2]=>660 [4,3,3,1,1]=>2970 [4,3,2,2,1]=>5940 [4,3,2,1,1,1]=>15840 [4,3,1,1,1,1,1]=>5544 [4,2,2,2,2]=>495 [4,2,2,2,1,1]=>7920 [4,2,2,1,1,1,1]=>13860 [4,2,1,1,1,1,1,1]=>5544 [4,1,1,1,1,1,1,1,1]=>495 [3,3,3,3]=>55 [3,3,3,2,1]=>1980 [3,3,3,1,1,1]=>2640 [3,3,2,2,2]=>990 [3,3,2,2,1,1]=>11880 [3,3,2,1,1,1,1]=>13860 [3,3,1,1,1,1,1,1]=>2772 [3,2,2,2,2,1]=>3960 [3,2,2,2,1,1,1]=>18480 [3,2,2,1,1,1,1,1]=>16632 [3,2,1,1,1,1,1,1,1]=>3960 [3,1,1,1,1,1,1,1,1,1]=>220 [2,2,2,2,2,2]=>132 [2,2,2,2,2,1,1]=>2772 [2,2,2,2,1,1,1,1]=>6930 [2,2,2,1,1,1,1,1,1]=>4620 [2,2,1,1,1,1,1,1,1,1]=>990 [2,1,1,1,1,1,1,1,1,1,1]=>66 [1,1,1,1,1,1,1,1,1,1,1,1]=>1 [13]=>1 [12,1]=>13 [10,3]=>13 [8,5]=>13 [7,6]=>13 [7,5,1]=>156 [7,4,2]=>156 [6,6,1]=>78 [6,4,2,1]=>1716 [5,5,3]=>78 [5,4,4]=>78 [5,4,3,1]=>1716 [5,4,2,2]=>858 [5,4,2,1,1]=>8580 [5,4,1,1,1,1]=>6435 [5,3,3,2]=>858 [5,3,3,1,1]=>4290 [5,3,2,2,1]=>8580 [5,3,2,1,1,1]=>25740 [4,4,4,1]=>286 [4,4,3,2]=>858 [4,4,3,1,1]=>4290 [4,4,2,2,1]=>4290 [4,3,3,3]=>286 [4,3,3,2,1]=>8580 [3,3,3,3,1]=>715 [3,3,3,2,2]=>1430 [3,3,2,2,2,1]=>12870 [3,2,2,2,2,2]=>1287 [2,2,2,2,2,2,1]=>1716 [1,1,1,1,1,1,1,1,1,1,1,1,1]=>1 [14]=>1 [13,1]=>14 [9,5]=>14 [8,5,1]=>182 [7,7]=>7 [7,5,2]=>182 [7,4,3]=>182 [6,6,2]=>91 [6,4,4]=>91 [6,2,2,2,2]=>1001 [5,5,4]=>91 [5,5,1,1,1,1]=>5005 [5,4,3,2]=>2184 [5,4,3,1,1]=>12012 [5,4,2,2,1]=>12012 [5,4,2,1,1,1]=>40040 [5,3,3,3]=>364 [5,3,3,2,1]=>12012 [5,2,2,2,2,1]=>10010 [4,4,4,2]=>364 [4,4,3,3]=>546 [4,4,3,2,1]=>12012 [4,3,2,2,2,1]=>40040 [3,3,3,3,2]=>1001 [3,3,3,3,1,1]=>5005 [3,3,2,2,2,2]=>5005 [2,2,2,2,2,2,2]=>429 [1,1,1,1,1,1,1,1,1,1,1,1,1,1]=>1 [15]=>1 [14,1]=>15 [9,5,1]=>210 [8,5,2]=>210 [7,5,3]=>210 [6,6,3]=>105 [6,5,4]=>210 [6,5,1,1,1,1]=>15015 [6,3,3,3]=>455 [6,2,2,2,2,1]=>15015 [5,5,5]=>35 [5,4,3,3]=>1365 [5,4,3,2,1]=>32760 [5,4,3,1,1,1]=>60060 [5,3,2,2,2,1]=>60060 [4,4,4,3]=>455 [4,4,4,1,1,1]=>10010 [4,3,3,3,2]=>5460 [3,3,3,3,3]=>273 [3,3,3,3,2,1]=>15015 [3,3,3,2,2,2]=>10010 [1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]=>1 [16]=>1 [15,1]=>16 [8,8]=>8 [8,5,3]=>240 [7,5,3,1]=>3360 [6,6,4]=>120 [5,5,3,3]=>840 [5,5,2,2,2]=>3640 [5,4,4,3]=>1680 [5,4,3,2,1,1]=>262080 [5,4,2,2,2,1]=>87360 [4,4,4,4]=>140 [4,4,4,2,2]=>3640 [4,4,3,3,2]=>10920 [4,3,3,3,3]=>1820 [4,3,3,3,2,1]=>87360 [3,3,3,3,2,2]=>10920 [2,2,2,2,2,2,2,2]=>1430 [17]=>1 [8,6,3]=>272 [7,5,3,2]=>4080 [6,5,3,3]=>2040 [6,5,2,2,2]=>9520 [6,4,4,3]=>2040 [6,4,4,1,1,1]=>61880 [6,3,3,3,2]=>9520 [6,3,3,3,1,1]=>61880 [5,5,4,3]=>2040 [5,5,4,1,1,1]=>61880 [5,5,2,2,2,1]=>61880 [5,4,4,4]=>680 [5,4,3,2,2,1]=>371280 [5,3,3,3,2,1]=>123760 [4,4,4,3,2]=>9520 [4,4,4,3,1,1]=>61880 [4,4,4,2,2,1]=>61880 [4,4,3,3,3]=>4760 [3,3,3,3,3,2]=>6188 [4,4,4,3,2,1]=>171360 [5,4,3,3,2,1]=>514080 [6,3,3,3,2,1]=>171360 [6,5,2,2,2,1]=>171360 [5,5,3,3,1,1]=>128520 [6,5,4,1,1,1]=>171360 [5,5,3,3,2]=>18360 [5,5,4,2,2]=>18360 [6,4,4,2,2]=>18360 [6,5,4,3]=>4896 [4,4,4,3,3]=>6120 [4,4,4,4,2]=>3060 [5,5,4,4]=>1224 [2,2,2,2,2,2,2,2,2]=>4862 [3,3,3,3,3,3]=>1428 [18]=>1 [6,6,6]=>51 [9,6,3]=>306 [8,6,4]=>306 [9,9]=>9 [5,4,4,3,2,1]=>697680 [5,5,3,3,2,1]=>348840 [5,5,4,2,2,1]=>348840 [6,4,4,2,2,1]=>348840 [5,5,4,3,1,1]=>348840 [6,4,4,3,1,1]=>348840 [6,5,3,3,1,1]=>348840 [5,5,4,3,2]=>46512 [6,4,4,3,2]=>46512 [6,5,3,3,2]=>46512 [6,5,4,2,2]=>46512 [6,5,4,3,1]=>93024 [6,5,4,1,1,1,1]=>813960 [4,4,4,4,3]=>3876 [4,3,3,3,3,3]=>11628 [19]=>1 [9,6,4]=>342 [8,5,4,2]=>5814 [8,5,5,1]=>2907 [5,5,4,3,2,1]=>930240 [6,4,4,3,2,1]=>930240 [6,5,3,3,2,1]=>930240 [6,5,4,2,2,1]=>930240 [6,5,4,3,1,1]=>930240 [6,5,4,3,2]=>116280 [6,5,2,2,2,2,1]=>1162800 [6,5,4,2,1,1,1]=>4651200 [7,5,4,3,1]=>116280 [3,3,3,3,3,3,2]=>38760 [4,4,3,3,3,3]=>38760 [4,4,4,4,4]=>969 [5,5,5,5]=>285 [20]=>1 [8,6,4,2]=>6840 [10,6,4]=>380 [10,7,3]=>380 [9,7,4]=>380 [9,5,5,1]=>3420 [6,5,4,3,2,1]=>2441880 [6,3,3,3,3,2,1]=>1627920 [6,5,3,2,2,2,1]=>6511680 [6,5,4,3,1,1,1]=>6511680 [3,3,3,3,3,3,3]=>7752 [4,4,4,3,3,3]=>67830 [21]=>1 [11,7,3]=>420 [4,4,4,4,3,2,1]=>2238390 [6,4,3,3,3,2,1]=>8953560 [6,5,4,2,2,2,1]=>8953560 [6,5,4,3,2,1,1]=>26860680 [4,4,4,4,3,3]=>65835 [9,6,4,3]=>9240 [5,4,4,4,3,2,1]=>12113640 [6,5,3,3,3,2,1]=>12113640 [6,5,4,3,2,2,1]=>36340920 [9,6,5,3]=>10626 [8,6,5,3,1]=>212520 [6,4,4,4,3,2,1]=>16151520 [6,5,4,3,3,2,1]=>48454560 [3,3,3,3,3,3,3,3]=>43263 [4,4,4,4,4,4]=>7084 [11,7,5,1]=>12144 [9,7,5,3]=>12144 [8,8,8]=>92 [5,5,5,4,3,2,1]=>21252000 [6,5,4,4,3,2,1]=>63756000 [9,7,5,3,1]=>303600 [10,7,5,3]=>13800 [6,5,5,4,3,2,1]=>82882800 [9,7,5,4,1]=>358800 [6,6,5,4,3,2,1]=>106563600 [7,6,5,4,3,2]=>9687600 [3,3,3,3,3,3,3,3,3]=>246675 [7,6,5,4,3,2,1]=>271252800 [7,6,5,4,3,1,1,1]=>994593600 [10,7,6,4,1]=>491400 [9,7,6,4,2]=>491400 [10,8,5,4,1]=>491400 [7,6,5,4,2,2,2,1]=>1311055200 [10,8,6,4,1]=>570024 [9,7,5,5,3,1]=>8550360 [7,6,5,3,3,3,2,1]=>1710072000 [11,8,6,4,1]=>657720 [10,8,6,4,2]=>657720 [11,8,6,5,1]=>755160 [4,4,4,4,4,4,4,4]=>420732 [12,9,7,5,1]=>1113024 [13,9,7,5,1]=>1256640 [11,9,7,5,3,1]=>45239040 [11,8,7,5,4,1]=>45239040 [11,9,7,5,5,3]=>39480480 [11,9,7,7,5,3,3]=>1466110800
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Description
The Kreweras number of an integer partition.
This is defined for $\lambda \vdash n$ with $k$ parts as
$$\frac{1}{n+1}\binom{n+1}{n+1-k,\mu_1(\lambda),\ldots,\mu_n(\lambda)}$$
where $\mu_j(\lambda)$ denotes the number of parts of $\lambda$ equal to $j$, see [1]. This formula indeed counts the number of noncrossing set partitions where the ordered block sizes are the partition $\lambda$.
These numbers refine the Narayana numbers $N(n,k) = \frac{1}{k}\binom{n-1}{k-1}\binom{n}{k-1}$ and thus sum up to the Catalan numbers $\frac{1}{n+1}\binom{2n}{n}$.
References
[1] Reiner, V., Sommers, E. Weyl group $q$-Kreweras numbers and cyclic sieving arXiv:1605.09172
[2] Coefficients T(j, k) of a partition transform for Lagrange compositional inversion of a function or generating series in terms of the coefficients of the power series for its reciprocal. Enumeration of noncrossing partitions and primitive parking functions. T(n,k) for n >= 1 and 1 <= k <= A000041(n-1), an irregular triangle read by rows. OEIS:A134264
Code
def statistic(la):
    la = list(la)
    n = sum(la)
    k = len(la)
    multi = [n+1-k]+[ la.count(j) for j in [1..n] ]
    return multinomial(multi)/(n+1)
Created
May 31, 2016 at 14:57 by Christian Stump
Updated
Jul 29, 2025 at 14:16 by Martin Rubey