A color pattern consists of a string of n letters, with each letter standing for a separate color. If we permute the letters, the resultant pattern is equivalent. Thus ABC, ACB, BAC, BCA, CAB, and CBA are equivalent color patterns. The number of nonequivalent color patterns for a string of n colors containing exactly k different colors is the Stirling subset number S2(n,k). For example, if n=3 and k=2, we have AAB, ABA, and ABB. Note that none of these is a permutation of another. A color pattern is achiral if its reverse (or a permutation of the colors thereof) is the same as the original. For example, if n=5 and k=3, there are five achiral color patterns: AABCC, ABACA, ABBBC, ABCAB, and ABCBA. Results for odd n appear in OEIS A140735; for even n in OEIS A293181. I have a recursive method to determine the number of achiral color patterns, but I am wondering if there is a closed-form solution.

A color pattern consists of a string of n letters, with each letter standing for a separate color. If we permute the letters, the resultant pattern is equivalent. Thus ABC, ACB, BAC, BCA, CAB, and CBA are equivalent color patterns. The number of nonequivalent color patterns for a string of n colors containing exactly k different colors is the Stirling subset number S2(n,k). For example, if n=3 and k=2, we have AAB, ABA, and ABB. Note that none of these is a permutation of another. A color pattern is achiral if its reverse (or a permutation of the colors thereof) is the same as the original. For example, if n=5 and k=3, there are five achiral color patterns: AABCC, ABACA, ABBBC, ABCAB, and ABCBA. Results for odd n appear in OEIS A140735; for even n in OEIS A293181. I have a recursive method to determine the number of achiral color patterns, but I am wondering if there is a closed-form solution.

A color pattern consists of a string of n letters, with each letter standing for a separate color. If we permute the letters, the resultant pattern is equivalent. Thus ABC, ACB, BAC, BCA, CAB, and CBA are equivalent color patterns. The number of nonequivalent color patterns for a string of n colors containing exactly k different colors is the Stirling subset number S2(n,k). For example, if n=3 and k=2, we have AAB, ABA, and ABB. Note that none of these is a permutation of another. A color pattern is achiral if its reverse (or a permutation of the colors thereof) is the same as the original. For example, if n=5 and k=3, there are five achiral color patterns: AABCC, ABACA, ABBBC, ABCAB, and ABCBA. Results for odd n appear in OEIS A140375; for even n in OEIS A293181. I have a recursive method to determine the number of achiral color patterns, but I am wondering if there is a closed-form solution.

A color pattern consists of a string of n letters, with each letter standing for a separate color. If we permute the letters, the resultant pattern is equivalent. Thus ABC, ACB, BAC, BCA, CAB, and CBA are equivalent color patterns. The number of nonequivalent color patterns for a string of n colors containing exactly k different colors is the Stirling subset number S2(n,k). For example, if n=3 and k=2, we have AAB, ABA, and ABB. Note that none of these is a permutation of another. A color pattern is achiral if its reverse (or a permutation of the colors thereof) is the same as the original. For example, if n=5 and k=3, there are five achiral color patterns: AABCC, ABACA, ABBBC, ABCAB, and ABCBA. Results for odd n appear in OEIS A140735; for even n in OEIS A293181. I have a recursive method to determine the number of achiral color patterns, but I am wondering if there is a closed-form solution.