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User:Olivier Gérard/IntPartitions
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Sequences which are of the form : "sum of f(p(i)) over all integer partitions of n".
| OEIS anum | OEIS values | Formula for f |
|---|---|---|
| A373243 | 0, 0, 1, 1, 2, 5, 6, 11, 18, 27 | Length Union - Union Length/@ Split |
Sequences with linear row sums equal to number of partitions
| OEIS anum | OEIS values | Type of sum | Name |
|---|---|---|---|
| A050314 | {1, 0, 1, 1, 0, 1, 0, 1, 0, 2, 2, 0, 2, 0, 1} | A000041(n) | Triangle: a(n,k) = number of partitions of n whose xor-sum is k. |
| A072233 | {1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1} | A000041(n) | Square array T(n,k) read by antidiagonals giving number of ways to distribute n indistinguishable objects in k indistinguishable containers; containers may be left empty. |
| A096651 | {1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1} | A000041(n) | Lower triangular matrix T, read by rows, such that the row sums of T^n form the n-dimensional partitions. |
| A097567 | {1, 1, 0, 0, 0, 2, 1, 0, 2, 0, 3, 0, 0, 0, 2} | A000041(n) | T(n,k)= count of partitions p such that Abs( Odd(p)-Odd(p') ) = k, where p' is the transpose of p and Odd(p) counts the odd elements in p. Related to Stanley's 'f'. |
| A103919 | {1, 0, 1, 1, 0, 1, 0, 2, 0, 1, 2, 0, 2, 0, 1} | A000041(n) | Triangle of numbers of partitions of n with total number of odd parts equal to k from {0,...,n}. |
| A113685 | {1, 0, 1, 1, 0, 1, 0, 1, 0, 2, 2, 0, 1, 0, 2} | A000041(n) | Triangular array read by rows: T(n,k) is the number of partitions of n in which sum of odd parts is k, for k=0,1,...,n; n>=0. |
| A113686 | {1, 1, 0, 1, 0, 1, 2, 0, 1, 0, 2, 0, 1, 0, 2} | A000041(n) | Triangular array T(n,k)=number of partitions of n in which sum of even parts is k, for k=0,1,...n; n>=0. |
| A116598 | {1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1} | A000041(n) | Triangle read by rows: T(n,k) is the number of partitions of n having exactly k parts equal to 1 (n>=0, 0<=k<=n). |
| A118198 | {1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 2, 0, 1} | A000041(n) | Triangle read by rows: T(n,k) is the number of partitions of n having k parts equal to the size of the Durfee square (0<=k<=n). |
| A137586 | {1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 1, 0, 0, 1} | A000041(n) | Triangle read by rows: A026794 * A054525. |
| A176202 | {1, 0, 1, 0, 0, 2, 1, 0, 0, 2, 0, 1, 0, 0, 4} | A000041(n) | Convolution triangle, row sums = A000041. M = A000041 in each column with two interleaved zeros; Q = A000726 diagonalized with the rest zeros. A176202 = M*Q. |
| A209354 | {1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1} | A000041(n) | Triangular array: T(n,k) = number of partitions of n for which (maximal term)-(minimal term)=k, if 0<=k<n, and T(n,n)=1. |
| A238353 | {1, 1, 0, 2, 0, 0, 2, 1, 0, 0, 3, 1, 1, 0, 0} | A000041(n) | Triangle T(n,k) read by rows: T(n,k) is the number of partitions of n (as weakly ascending list of parts) with maximal ascent k, n >= 0, 0 <= k <= n. |
| A238354 | {1, 1, 0, 2, 0, 0, 2, 1, 0, 0, 4, 0, 1, 0, 0} | A000041(n) | Triangle T(n,k) read by rows: T(n,k) is the number of partitions of n (as weakly ascending list of parts) with minimal ascent k, n >= 0, 0 <= k <= n. |
| A243978 | {1, 0, 1, 0, 1, 1, 0, 2, 0, 1, 0, 3, 1, 0, 1} | A000041(n) | Triangle T(n,k), n>=0, 0<=k<=n, read by rows: T(n,k) is the number of partitions of n where the minimal multiplicity of any part is k. |
| A263233 | {1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1} | A000041(n) | Triangle read by rows: T(n,k) is the number of partitions of n having k perfect square parts (0<=k<=n). |
| A263234 | {1, 0, 1, 1, 0, 1, 0, 2, 0, 1, 2, 0, 2, 0, 1} | A000041(n) | Triangle read by rows: T(n,k) is the number of partitions of n having k triangular number parts (0<=k<=n). |
| A264391 | {1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1, 0, 1} | A000041(n) | Triangle read by rows: T(n,k) is the number of partitions of n having k perfect cube parts (0<=k<=n). |
| A264394 | {1, 0, 1, 1, 0, 1, 0, 2, 0, 1, 2, 0, 2, 0, 1} | A000041(n) | Triangle read by rows: T(n,k) is the number of partitions of n having k Mersenne number parts (0<=k<=n). |
| A264403 | {1, 0, 1, 1, 0, 1, 1, 0, 0, 2, 2, 0, 1, 0, 2} | A000041(n) | Triangle read by rows: T(n,k) is the number of partitions of n in which the sum of the parts of multiplicity 1 is equal to k (0<=k<=n). |
| A264405 | {1, 1, 0, 1, 0, 1, 2, 0, 0, 1, 2, 0, 2, 0, 1} | A000041(n) | Triangle read by rows: T(n,k) is the number of integer partitions of n having k repeated parts (each occurrence is counted). |
| A276422 | {1, 0, 1, 2, 0, 0, 1, 1, 0, 1, 4, 0, 0, 0, 1} | A000041(n) | Triangle read by rows: T(n,k) is the number of partitions of n for which the sum of its odd singletons is k (0<=k<=n). A singleton in a partition is a part that occurs exactly once. |
| A276424 | {1, 1, 0, 1, 0, 1, 2, 0, 1, 0, 3, 0, 1, 0, 1} | A000041(n) | Triangle read by rows: T(n,k) is the number of partitions of n for which the sum of its even singletons is k (0<=k<=n). A singleton in a partition is a part that occurs exactly once. |
| A307431 | {1, 0, 1, 0, 1, 1, 0, 1, 0, 2, 0, 1, 1, 2, 1} | A000041(n) | Number T(n,k) of partitions of n into parts whose bitwise OR equals k; triangle T(n,k), n>=0, 0<=k<=n, read by rows. |
| A307432 | {1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1} | A000041(n) | Number T(n,k) of partitions of n into parts whose bitwise AND equals k; triangle T(n,k), n>=0, 0<=k<=n, read by rows. |
| A308151 | {1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1} | A000041(n) | Triangular array: each row partitions the partitions of n into n parts; of which the k-th part is the number of partitions having stay number k-1; see Comments. |
| A322393 | {1, 0, 1, 1, 1, 0, 2, 1, 0, 0, 3, 1, 1, 0, 0} | A000041(n) | Regular triangle read by rows where T(n,k) is the number of integer partitions of n with edge-connectivity k, for 0 <= k <= n. |
| A325165 | {1, 0, 1, 0, 0, 2, 0, 0, 0, 3, 0, 1, 0, 0, 4} | A000041(n) | Regular triangle read by rows where T(n,k) is the number of integer partitions of n whose inner lining partition has last (smallest) part equal to k. |
| A325188 | {1, 0, 1, 0, 2, 0, 0, 2, 1, 0, 0, 2, 3, 0, 0} | A000041(n) | Regular triangle read by rows where T(n,k) is the number of integer partitions of n with origin-to-boundary graph-distance equal to k. |
| A325189 | {1, 0, 1, 0, 0, 2, 0, 0, 1, 2, 0, 0, 0, 3, 2} | A000041(n) | Regular triangle read by rows where T(n,k) is the number of integer partitions of n with maximum origin-to-boundary graph-distance equal to k. |
| A325192 | {1, 1, 0, 0, 2, 0, 0, 1, 2, 0, 1, 0, 2, 2, 0} | A000041(n) | Regular triangle read by rows where T(n,k) is the number of integer partitions of n such that the difference between the length of the minimal square containing and the maximal square contained in the Young diagram is k. |
| A325200 | {1, 1, 0, 0, 2, 0, 1, 0, 2, 0, 0, 3, 0, 2, 0} | A000041(n) | Regular triangle read by rows where T(n,k) is the number of integer partitions of n such that the difference between the length of the minimal triangular partition containing and the maximal triangular partition contained in the Young diagram is k. |
| A325268 | {1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 3, 0, 1} | A000041(n) | Triangle read by rows where T(n,k) is the number of integer partitions of n with omicron k. |
| A325280 | {1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1} | A000041(n) | Triangle read by rows where T(n,k) is the number of integer partitions of n with adjusted frequency depth k. |
| A325458 | {1, 0, 1, 0, 0, 2, 0, 0, 0, 3, 0, 0, 0, 1, 4} | A000041(n) | Triangle read by rows where T(n,k) is the number of integer partitions of n with largest hook of size k, i.e., with (largest part) + (number of parts) - 1 = k. |
| A325466 | {1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 3, 1, 0, 0} | A000041(n) | Triangle read by rows where T(n,k) is the number of reversed integer partitions of n with k distinct differences of any degree > 0. |
| A339737 | {1, 1, 0, 1, 1, 0, 2, 0, 1, 0, 2, 1, 1, 1, 0} | A000041(n) | Triangle read by rows where T(n,k) is the number of integer partitions of n with greatest gap k. |
| A344612 | {1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1} | A000041(n) | Triangle read by rows where T(n,k) is the number of integer partitions of n with reverse-alternating sum k ranging from -n to n in steps of 2. |
| A353315 | {1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 2, 1, 0, 1} | A000041(n) | Triangle read by rows where T(n,k) is the number of integer partitions of n with k parts on or below the diagonal (weak non-excedances). |
| A353836 | {1, 0, 1, 0, 2, 0, 0, 2, 1, 0, 0, 4, 1, 0, 0} | A000041(n) | Triangle read by rows where T(n,k) is the number of integer partitions of n with k distinct run-sums. |
| A353846 | {1, 0, 1, 0, 1, 1, 0, 2, 1, 0, 0, 2, 2, 1, 0} | A000041(n) | Triangle read by rows where T(n,k) is the number of integer partitions of n with partition run-sum trajectory of length k. |
| A357637 | {1, 0, 1, 0, 0, 2, 0, 0, 1, 2, 0, 0, 1, 1, 3} | A000041(n) | Triangle read by rows where T(n,k) is the number of integer partitions of n with half-alternating sum k, where k ranges from -n to n in steps of 2. |
| A357638 | {1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 0, 3, 1, 1} | A000041(n) | Triangle read by rows where T(n,k) is the number of integer partitions of n with skew-alternating sum k, where k ranges from -n to n in steps of 2. |
| A357704 | {1, 0, 1, 0, 0, 2, 0, 0, 1, 2, 0, 0, 2, 0, 3} | A000041(n) | Triangle read by rows where T(n,k) is the number of reversed integer partitions of n with half-alternating sum k, where k ranges from -n to n in steps of 2. |
| A357705 | {1, 0, 1, 0, 1, 1, 0, 2, 0, 1, 0, 2, 2, 0, 1} | A000041(n) | Triangle read by rows where T(n,k) is the number of reversed integer partitions of n with skew-alternating sum k, where k ranges from -n to n in steps of 2. |
| A360672 | {1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, 3, 1, 0} | A000041(n) | Triangle read by rows where T(n,k) is the number of integer partitions of n whose left half (exclusive) sums to k, where k ranges from 0 to n. |
| A360675 | {1, 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 2, 2, 0, 0} | A000041(n) | Triangle read by rows where T(n,k) is the number of integer partitions of n whose right half (exclusive) sums to k, where k ranges from 0 to n. |
| A363946 | {1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 3, 0, 1} | A000041(n) | Triangle read by rows where T(n,k) is the number of integer partitions of n with high mean k. |
| A363952 | {1, 0, 1, 0, 1, 1, 0, 2, 0, 1, 0, 3, 1, 0, 1} | A000041(n) | Number of integer partitions of n with low mode k. |
| A363953 | {1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 2, 1, 1, 1} | A000041(n) | Number of integer partitions of n with high mode k. |
| A364060 | {1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 2, 2, 0, 1} | A000041(n) | Triangle read by rows where T(n,k) is the number of integer partitions of n with rounded mean k. |
| A364916 | {1, 1, 0, 1, 1, 0, 2, 0, 1, 0, 2, 1, 1, 1, 0} | A000041(n) | Array read by anti-diagonals downwards where A(n,k) is the number of ways to write n as a nonnegative linear combination of the parts of a strict integer partition of k. |
| A365676 | {1, 0, 1, 0, 2, 0, 0, 2, 1, 0, 0, 3, 2, 0, 0} | A000041(n) | Triangle read by rows: T(n, k) is the number of partitions of n having exactly k distinct parts, for 0 <= k <= n. |
| A365921 | {1, 1, 0, 1, 1, 0, 2, 0, 1, 0, 2, 0, 1, 2, 0} | A000041(n) | Triangle read by rows where T(n,k) is the number of integer partitions y of n such that k is the greatest member of {0..n} that is not the sum of any nonempty submultiset of y. |
| A365923 | {1, 1, 0, 1, 1, 0, 2, 0, 1, 0, 2, 1, 1, 1, 0} | A000041(n) | Triangle read by rows where T(n,k) is the number of integer partitions of n with exactly k distinct non-subset-sums. |
| A367582 | {1, 0, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 2, 1, 1} | A000041(n) | Triangle read by rows where T(n,k) is the number of integer partitions of n whose multiset multiplicity kernel (in which each multiplicity becomes the least element of that multiplicity), sums to k. |
| A008284 | {1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1} | A000041(n+1) | Triangle of partition numbers: T(n,k) = number of partitions of n in which the greatest part is k, 1 <= k <= n. Also number of partitions of n into k positive parts, 1 <= k <= n. |
| A026794 | {1, 1, 1, 2, 0, 1, 3, 1, 0, 1, 5, 1, 0, 0, 1} | A000041(n+1) | Triangular array T read by rows: T(n,k) = number of partitions of n in which least part is k, 1<=k<=n. |
| A049597 | {1, 0, 2, 0, 0, 3, 0, 0, 1, 4, 0, 0, 0, 2, 5} | A000041(n+1) | Triangular array T(n,k) in which k-th column gives coefficients of sum of Gaussian polynomials [k,m] for m=0..k. |
| A052249 | {1, 1, 1, 0, 2, 1, 0, 1, 3, 1, 0, 0, 2, 4, 1} | A000041(n+1) | Triangle T(n,k) (n >= 1, k >= 1) giving dimension of bigrading of Connes-Moscovici noncocommutative algebra. |
| A058398 | {1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1} | A000041(n+1) | Partition triangle A008284 read from right to left. |
| A091602 | {1, 1, 1, 2, 0, 1, 2, 2, 0, 1, 3, 2, 1, 0, 1} | A000041(n+1) | Triangle: T(n,k) is the number of partitions of n such that some part is repeated k times and no part is repeated more than k times. |
| A096144 | {1, 1, 1, 2, 0, 1, 2, 2, 0, 1, 4, 1, 1, 0, 1} | A000041(n+1) | Triangle T(n,k) = number of partitions of n in which the least part occurs exactly k times, k=1..n. |
| A096771 | {1, 0, 2, 0, 1, 2, 0, 1, 2, 2, 0, 0, 3, 2, 2} | A000041(n+1) | Triangle read by rows: T(n,m) is the number of partitions of n that (just) fit inside an m X m box, but not in an (m-1) X (m-1) box. Partitions of n with Max(max part, length) = m. |
| A097364 | {1, 2, 0, 2, 1, 0, 3, 1, 1, 0, 2, 3, 1, 1, 0} | A000041(n+1) | Triangle read by rows, 0 <= k < n: T(n,k) = number of partitions of n such that the differences between greatest and smallest parts are k. |
| A114087 | {1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1} | A000041(n+1) | Triangle read by rows: T(n,k) is the number of partitions of n whose tails below their Durfee squares have size k (n>=1; 0<=k<=n-1). |
| A114088 | {1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1} | A000041(n+1) | Triangle read by rows: T(n,k) is number of partitions of n whose tail below its Durfee square has k parts (n >= 1; 0 <= k <= n-1). |
| A115723 | {1, 0, 2, 0, 1, 2, 0, 0, 2, 3, 0, 0, 1, 4, 2} | A000041(n+1) | Table of partitions of n with maximum rectangle k. |
| A116861 | {1, 1, 1, 1, 0, 2, 1, 1, 1, 2, 1, 0, 2, 1, 3} | A000041(n+1) | Triangle read by rows: T(n,k) is the number of partitions of n such that the sum of the parts, counted without multiplicities, is equal to k (n>=1, k>=1). |
| A124943 | {1, 1, 1, 2, 0, 1, 3, 1, 0, 1, 4, 2, 0, 0, 1} | A000041(n+1) | Table read by rows: number of partitions of n with k as low median. |
| A124944 | {1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1, 1} | A000041(n+1) | Table, number of partitions of n with k as high median. |
| A130162 | {1, 1, 1, 1, 0, 2, 1, 1, 0, 3, 1, 0, 0, 0, 6} | A000041(n+1) | Triangle read by rows: A051731 * A000837 as a diagonalized matrix. |
| A132825 | {1, 0, 2, 0, 0, 3, 0, 0, 0, 5, 0, 0, 0, 0, 7} | A000041(n+1) | Triangle read by rows: zeros except for right border which are the partition numbers A000041. |
| A133121 | {1, 1, 1, 2, 0, 1, 2, 2, 0, 1, 3, 2, 1, 0, 1} | A000041(n+1) | Triangle T(n,k) read by rows = number of partitions of n such that number of parts minus number of distinct parts is equal to k, k = 0..n-1. |
| A134979 | {1, 0, 2, 0, 1, 2, 0, 1, 1, 3, 0, 0, 3, 2, 2} | A000041(n+1) | Triangle read by rows: T(n,k) = number of partitions of n where the maximum number of objects in partitions of any given size is k. |
| A135486 | {1, 0, 2, 1, 0, 2, 1, 1, 0, 3, 5, 0, 0, 0, 2} | A000041(n+1) | Triangle read by rows: T(n,k) = number of partitions of n having k-fold symmetry, cf. A085436. |
| A168532 | {1, 1, 1, 2, 0, 1, 3, 1, 0, 1, 6, 0, 0, 0, 1} | A000041(n+1) | Triangle read by rows, A054525 * A168021. |
| A174067 | {1, 1, 1, 2, 0, 1, 3, 1, 0, 1, 4, 1, 1, 0, 1} | A000041(n+1) | Triangle, row sums = A000041 starting (1, 2, 3, 5, 7, ...); derived from finite differences of p(x) = A(x)*A(x^2) = B(x)*B(x^3) = C(x)*C(x^4) = ... |
| A175010 | {1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3} | A000041(n+1) | Triangle generated from INVERT transforms of variants of A080995. |
| A194799 | {1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1} | A000041(n+1) | Triangle read by rows: T(n,k) = number of partitions of n that are formed by k sections, k >= 1. |
| A218907 | {1, 2, 0, 2, 0, 1, 2, 0, 2, 1, 2, 0, 2, 2, 1} | A000041(n+1) | Triangle, read by rows, of integer partitions of n by kernel size k. |
| A230025 | {1, 0, 2, 1, 0, 2, 1, 2, 0, 2, 1, 2, 2, 0, 2} | A000041(n+1) | Triangular array: t(n, k) = number of occurrences of k as the number of outliers in all the partitions of n. |
| A237513 | {1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1} | A000041(n+1) | T(n,k) = number of maximal horizontal rectangles that contain the Durfee square for partitions of n that consist of k nodes, 1 <= k <= n; triangular array read by rows. |
| A265247 | {1, 2, 0, 2, 0, 1, 3, 0, 1, 1, 2, 0, 2, 2, 1} | A000041(n+1) | Triangle read by rows: T(n,k) is the number of partitions of n in which the 2nd smallest part is k when the partition has at least 2 distinct parts and 0 otherwise; (n>=1, 0 <= k <= n). |
| A268189 | {1, 2, 0, 2, 0, 1, 3, 0, 1, 1, 2, 0, 1, 2, 2} | A000041(n+1) | Triangle read by rows: T(n,k) is the number of partitions of n for which the sum of the parts larger than the smallest part is k (n>=1, 0<=k<=n-1). |
| A279044 | {1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 2, 1, 1} | A000041(n+1) | Irregular triangular array: T(n,i) = number of partitions of n having crossover part k; see Comments. |
| A303810 | {1, 1, 1, 1, 0, 2, 1, 0, 1, 3, 1, 0, 0, 1, 5} | A000041(n+1) | Mirror image of the triangle A026794. |
| A329746 | {1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 0} | A000041(n+1) | Triangle read by rows where T(n,k) is the number of integer partitions of n > 0 with runs-resistance k, 0 <= k <= n - 1. |
| A330374 | {1, 0, 2, 1, 0, 2, 1, 2, 0, 2, 1, 2, 2, 0, 2} | A000041(n+1) | Triangle read by rows: T(n,k) is the number of partitions of n whose absolute value of Dyson's rank is equal to k, with 0 <= k < n. |
| A341418 | {1, 1, 1, 0, 2, 1, 0, 1, 3, 1, -1, 0, 3, 4, 1} | A000041(n+1) | Triangle read by rows: T(n, m) gives the sum of the weights of weighted compositions of n with m parts from generalized pentagonal numbers {A001318(k)}_{k>=1}. |
| A365658 | {1, 1, 1, 1, 0, 2, 1, 1, 1, 2, 1, 0, 2, 0, 4} | A000041(n+1) | Triangle read by rows where T(n,k) is the number of integer partitions of n with k distinct possible sums of nonempty submultisets. |
| A366745 | {1, 0, 2, 1, 0, 2, 1, 2, 0, 2, 1, 2, 2, 0, 2} | A000041(n+1) | Triangular array, read by rows: T(n,k) = number of partitions p of n such that d(p, p') = 2k, where p' = contraconjugate of p, and d is the distance function defined in A366156. |
